Try to start Firefox in safe mode.
https://support.mozilla.org/en-US/kb/troubleshoot-firefox-issues-using-safe-mode

Do you still see the problem?

Surely you jest!
Way too many extensions! That's just asking for trouble.
Plus with the new Australis interface in Firefox 29, many of the extensions that are disabled due to incompatibility (Inactive) will probably never be updated. This is when "we" lose many add-on developers, when there are major changes in Firefox and they don't want to start from scratch to replicate their add-on.
Plus. it looks like you have too many add-ons that do the same thing or have similar features, which in itself can produce problems.

IMO, you're gonna waste more time trying to fix that Profile (with little chance of success), than if you just "bit the bullet" and started from scratch. Been there, done that a few dozen times with Firefox in the early days. Profiles don't last forever; major changes to Firefox or running too many extensions will create hiccups or faulty interactions that are all but impossible for even the developers to figure out.

Extensions

    Adblock Plus 2.5.1 ({d10d0bf8-f5b5-c8b4-a8b2-2b9879e08c5d})
    Add-on Update Checker 2.5 ({891f0410-aaa2-11e0-9f1c-0800200c9a66})
    Addons Recent Updates 0.1.6 (addonsRecentUpdates@infocatcher)
    AutoCopy 2 1.2.8.4 (autocopy2@teo.pl)
    Cache Status 0.7.10 (cache@status.org)
    Classic Bookmarks Button 0.9 (ClassicBookmarksButton@ArisT2Noia4dev)
    Cleanest Addon Manager 7.0 (cam@sdrocking.com)
    Context Search 0.6.2 ({902D2C4A-457A-4EF9-AD43-7014562929FF})
    Context Search X 0.4.6.13 (contextsearch2@lwz.addons.mozilla.org)
    Customizations for Adblock Plus 1.0.3 (customization@adblockplus.org)
    Download Panel Tweaker 0.2.0 (downloadPanelTweaker@infocatcher)
    DownThemAll! 2.0.16 ({DDC359D1-844A-42a7-9AA1-88A850A938A8})
    Element Hiding Helper for Adblock Plus 1.2.3 (elemhidehelper@adblockplus.org)
    EventBug 0.1b10 (eventbug@getfirebug.com)
    feedly 16.0.528 (feedly@devhd)
    Firebug 1.12.7 (firebug@software.joehewitt.com)
    FireGestures 1.8.4 (firegestures@xuldev.org)
    Flashblock 1.5.17 ({3d7eb24f-2740-49df-8937-200b1cc08f8a})
    Grab and Drag 3.2.5.2 ({477c4c36-24eb-11da-94d4-00e08161165f})
    Greasemonkey 1.15 ({e4a8a97b-f2ed-450b-b12d-ee082ba24781})
    gTranslate 0.9 ({aff87fa2-a58e-4edd-b852-0a20203c1e17})
    InlineDisposition Reloaded 1.0.0.0 ({914d3a0b-6982-4e58-9a60-2e5d6cf6b09b})
    iReader 1.0.7 (ireader@samabox.com)
    Lazarus: Form Recovery 2.3 (lazarus@interclue.com)
    Link Alert 1.0.2 (linkalert.conlan@addons.mozilla.com)
    Memory Menu 0.3 (memorymenu@shivamdutt.net)
    Offline Switch 0.5.1 (jid1-2fubm22eqCjBDQ@jetpack)
    PDF Viewer 0.8.1280 (uriloader@pdf.js)
    Permanent List-all-tabs Button 1.0 (listalltabs@sdrocking.com)
    Places Maintenance 1.3 (places-maintenance@bonardo.net)
    Pocket 3.0.4 (isreaditlater@ideashower.com)
    Print Edit 11.4 (printedit@DW-dev)
    Quick Translator 1.0 ({5C655500-E712-41e7-9349-CE462F844B19})
    Reddit Enhancement Suite 4.3.1.2 (jid1-xUfzOsOFlzSOXg@jetpack)
    Restartless Restart 9 (restartless.restart@erikvold.com)
    RSS Icon in url bar 1.3 (rssicon.vaka@gmail.com)
    ScrapBook 1.5.8 ({53A03D43-5363-4669-8190-99061B2DEBA5})
    Search by Image for Google 1.2.0 ({ab4b5718-3998-4a2c-91ae-18a7c2db513e})
    Session Manager 0.8.1.0 ({1280606b-2510-4fe0-97ef-9b5a22eafe30})
    SPDY indicator 2.1 (spdyindicator@chengsun.github.com)
    Suspend Tab 0.1.2013111801 (suspendtab@piro.sakura.ne.jp)
    Tab Counter 1.9.9 (tabcounter@morac)
    Troubleshooter 1.1a (troubleshooter@mozilla.org)
    URL Dot 0.4.3 (urlDot@Chewybyte.net)
    Vacuum Places Improved 1.2 (VacuumPlacesImproved@lultimouomo-gmail.com)
    Video Resumer 1.1.3 (videoresumer@jetpack)
    VKontakte.ru Downloader 0.3.7.0 (vk@sergeykolosov.mp)
    vkPatch 6.1.7 (vkpatch@pochemuto.org)
    YouTube Center 2.1.0 (jid1-cwbvBTE216jjpg@jetpack)
    "Yandex Elements" 8.1.1 (yasearch@yandex.ru) (Inactive)
    Скачивалка музыки для ВКонтакте.ру и Яндекс.Музыки 0.7.5 (vmpeg@vmpeg.ru) (Inactive)
    about:addons-memory 8 (about-addons-memory@tn123.org) (Inactive)
    Autocopy 1.0.8 ({0FED7D55-65D4-47b6-A6DE-9A4ADB55355F}) (Inactive)
    Cheevos 1.5 (jid1-bpzDizt9E1R7nw@jetpack) (Inactive)
    ChromEdit Plus 2.9.10 (chromeditplus@webdesigns.ms11.net) (Inactive)
    Classic Theme Restorer 1.1.7 (ClassicThemeRestorer@ArisT2Noia4dev) (Inactive)
    Context Search RG 1.1 (contextsearchRG@mozilla.org) (Inactive)
    Cookies Manager+ 1.5.2 ({bb6bc1bb-f824-4702-90cd-35e2fb24f25d}) (Inactive)
    Download Status Bar 8.1.5 ({6c28e999-e900-4635-a39d-b1ec90ba0c0f}) (Inactive)
    Download Statusbar 0.9.10 ({D4DD63FA-01E4-46a7-B6B1-EDAB7D6AD389}) (Inactive)
    Downloads in Tab 1.00 (downintab@max.max) (Inactive)
    Easy Context Search 1.2 (easycontextsearch_manishjain9@hotmail.com) (Inactive)
    Easy YouTube Video Downloader 7.0 ({c0c9a2c7-2e5c-4447-bc53-97718bc91e1b}) (Inactive)
    FB2 Reader 0.24 (fb2reader@clear.com.ua) (Inactive)
    FlipClock 0.82 ({cdd09450-7280-11de-8a39-0800200c9a66}) (Inactive)
    FXChromeMods 1.2.0 (jid0-zDuE7MQZjTEpOLHPvhw3GbDyhIg@jetpack) (Inactive)
    Hide My Ass Proxy Extension 1.2.7 (extension@hidemyass.com) (Inactive)
    InlineDisposition 1.0.2.4 ({123647d5-da43-4344-bfe2-fc093bdf8f5e}) (Inactive)
    JSON-handle 0.2.2.1 (jsonhandle@gmail.com) (Inactive)
    Light Switch 0.7 ({600452e8-6851-46db-80fd-fa571b2deaa7}) (Inactive)
    Lightbeam 1.0.9 (jid1-F9UJ2thwoAm5gQ@jetpack) (Inactive)
    Logitech Device Detection 1.24.0.9 (DeviceDetection@logitech.com) (Inactive)
    Logitech SetPoint 6.5 ({F003DA68-8256-4b37-A6C4-350FA04494DF}) (Inactive)
    MemChaser 0.5.2.1 (memchaser@quality.mozilla.org) (Inactive)
    Memory Restart 1.16 (memoryrestart@teamextension.com) (Inactive)
    Mozilla Archive Format 3.0.0.0 ({7f57cf46-4467-4c2d-adfa-0cba7c507e54}) (Inactive)
    music_detector 0.8.6 (jid0-zoF9h0SYUQ1AIJNme8UGyJxW3V0@jetpack) (Inactive)
    Nokia Drop 1.5.1 (jid1-uzLUl48X4QaYDA@jetpack) (Inactive)
    Organize Search Engines 1.7 (organize-search-engines@maltekraus.de) (Inactive)
    ProxTube - Unblock YouTube 1.5.5 (ich@maltegoetz.de) (Inactive)
    Reader 4.4 ({20068ab2-1901-4140-9f3c-81207d4dacc4}) (Inactive)
    Remove Temporarily 1.0.1 (removetemporarily@adblockplus.org) (Inactive)
    Restartless Context Search 1.2 (Contextsearch@yahoo.com) (Inactive)
    S3.Download Statusbar 2.04 (s3download@statusbar) (Inactive)
    SaveFrom.net helper 2.37 (helper@savefrom.net) (Inactive)
    Star-Button In Urlbar 1.0.4 (Starbuttoninurlbar@ArisT2Noia4dev) (Inactive)
    Stop YouTube Autoplay 1.0.3 (jid0-Pm0nbsggUvL00CBoW6YwCaqv8bk@jetpack) (Inactive)
    Suspend background tabs 1.0.1 (suspendbackgroundtabs@adblockplus.org) (Inactive)
    Test Pilot 1.2.3 (testpilot@labs.mozilla.com) (Inactive)
    TinEye Reverse Image Search 1.1 (tineye@ideeinc.com) (Inactive)
    TryAgain 3.4.6 ({992791ee-61dc-7b98-a8fd-dc49b7deeee9}) (Inactive)
    UnMHT 7.0.3 ({f759ca51-3a91-4dd1-ae78-9db5eee9ebf0}) (Inactive)
    User Agent Overrider 0.2.3 (useragentoverrider@qixinglu.com) (Inactive)
    Vimperator 3.8.2 (vimperator@mozdev.org) (Inactive)
    Visual Bookmarks 2.10.0 (vb@yandex.ru) (Inactive)
    Vkontakte photo 0.0.0.4 (89da06f4-402b-4cc7-bb6e-f73e60c0a844@pihel.jino.ru) (Inactive)
    Vkontakte tools 0.4.3.0 (ead2aeb4-f0e5-4ec5-a359-375e7f424100@chopix.ru) (Inactive)
    WOT 20131118 ({a0d7ccb3-214d-498b-b4aa-0e8fda9a7bf7}) (Inactive)
    Yet Another Context Search 1.3.1 (yacs@tcr.org.au) (Inactive)
    Youtube AutoPlay Stopper 1.0.7 (youtubeautoplaystopper@grizzlyape.com) (Inactive)
    YouTube tools 0.0.0.6 (cb08ed4b-a4ac-47f1-bb5a-06f268e361a1@pihel.jino.ru) (Inactive)
    YouTube™ Anywhere Player 2.8.6 ({c9d31470-81c6-4e3e-9a37-46eb9237ed3a}) (Inactive)

In case you are using "Clear history when Firefox closes":

• do not clear the Site Preferences

• Tools > Options > Privacy > Firefox will: "Use custom settings for history": [X] "Clear history when Firefox closes" > Settings
• https://support.mozilla.org/kb/Clear+Recent+History

Note that clearing "Site Preferences" clears all exceptions for cookies, images, pop-up windows, software installation, passwords, and other website specific data.

Clearing 'Site Preferences' (in 'Clear Recent History' or Ctrl + Shift + Del) even for the 'Last Hour' helped.

(permissions.sqlite contained empty table instead of default rows)

the-edmeister,

Don't worry, some exts with similar functions don't interfere (cus now it's not first versions of FF, there were troubles that days), they all use for their features its analogs haven't got, and overlapping functions are disabled. Other exts are disabled. As I figured out on the forums, disabled addons don't make much load to the Firefox. I use disabled exts as a list of helpful addons because it may be tough to remember useful addons for some situations.

Note that Firefox will still check if there are updates for disabled extensions.

Do you have to keep continuing to clear the Site Preferences or at least do this once per session or has it been enough to only do this once?

If you have to do this more often then it is possible that the permissions.sqlite file is corrupted or that you have other software (CCleaner, Advanced SystemCare) that is causing this issue.

It required only once to recreate correct permissions.sqlite. I restarted FF a few times, everything seems to be OK. I don't have CCleaner or any other kind of stuff.

May be it will be more effective to use collections on AMO, but I don't like its current design.

Did you already looked at the Australis appearance that will arrive in the next Firefox 29 release?

• https://support.mozilla.org/kb/common-questions-after-updating-to-new-firefox
• https://support.mozilla.org/kb/learn-more-about-the-design-of-new-firefox
• https://support.mozilla.org/kb/how-to-make-new-firefox-look-like-old-firefox
• https://support.mozilla.org/kb/what-happened-to-the-add-on-bar

Yes, I'm already on 29.0, like it specified in this question's Additional System Details but I see no relate to the current topic.