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 ([email protected])
    Cache Status 0.7.10 ([email protected])
    Classic Bookmarks Button 0.9 (ClassicBookmarksButton@ArisT2Noia4dev)
    Cleanest Addon Manager 7.0 ([email protected])
    Context Search 0.6.2 ({902D2C4A-457A-4EF9-AD43-7014562929FF})
    Context Search X 0.4.6.13 ([email protected])
    Customizations for Adblock Plus 1.0.3 ([email protected])
    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 ([email protected])
    EventBug 0.1b10 ([email protected])
    feedly 16.0.528 (feedly@devhd)
    Firebug 1.12.7 ([email protected])
    FireGestures 1.8.4 ([email protected])
    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 ([email protected])
    Lazarus: Form Recovery 2.3 ([email protected])
    Link Alert 1.0.2 ([email protected])
    Memory Menu 0.3 ([email protected])
    Offline Switch 0.5.1 (jid1-2fubm22eqCjBDQ@jetpack)
    PDF Viewer 0.8.1280 ([email protected])
    Permanent List-all-tabs Button 1.0 ([email protected])
    Places Maintenance 1.3 ([email protected])
    Pocket 3.0.4 ([email protected])
    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 ([email protected])
    RSS Icon in url bar 1.3 ([email protected])
    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 ([email protected])
    Suspend Tab 0.1.2013111801 ([email protected])
    Tab Counter 1.9.9 (tabcounter@morac)
    Troubleshooter 1.1a ([email protected])
    URL Dot 0.4.3 ([email protected])
    Vacuum Places Improved 1.2 (VacuumPlacesImproved@lultimouomo-gmail.com)
    Video Resumer 1.1.3 (videoresumer@jetpack)
    VKontakte.ru Downloader 0.3.7.0 ([email protected])
    vkPatch 6.1.7 ([email protected])
    YouTube Center 2.1.0 (jid1-cwbvBTE216jjpg@jetpack)
    "Yandex Elements" 8.1.1 ([email protected]) (Inactive)
    Скачивалка музыки для ВКонтакте.ру и Яндекс.Музыки 0.7.5 ([email protected]) (Inactive)
    about:addons-memory 8 ([email protected]) (Inactive)
    Autocopy 1.0.8 ({0FED7D55-65D4-47b6-A6DE-9A4ADB55355F}) (Inactive)
    Cheevos 1.5 (jid1-bpzDizt9E1R7nw@jetpack) (Inactive)
    ChromEdit Plus 2.9.10 ([email protected]) (Inactive)
    Classic Theme Restorer 1.1.7 (ClassicThemeRestorer@ArisT2Noia4dev) (Inactive)
    Context Search RG 1.1 ([email protected]) (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 ([email protected]) (Inactive)
    Easy Context Search 1.2 ([email protected]) (Inactive)
    Easy YouTube Video Downloader 7.0 ({c0c9a2c7-2e5c-4447-bc53-97718bc91e1b}) (Inactive)
    FB2 Reader 0.24 ([email protected]) (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 ([email protected]) (Inactive)
    InlineDisposition 1.0.2.4 ({123647d5-da43-4344-bfe2-fc093bdf8f5e}) (Inactive)
    JSON-handle 0.2.2.1 ([email protected]) (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 ([email protected]) (Inactive)
    Logitech SetPoint 6.5 ({F003DA68-8256-4b37-A6C4-350FA04494DF}) (Inactive)
    MemChaser 0.5.2.1 ([email protected]) (Inactive)
    Memory Restart 1.16 ([email protected]) (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 ([email protected]) (Inactive)
    ProxTube - Unblock YouTube 1.5.5 ([email protected]) (Inactive)
    Reader 4.4 ({20068ab2-1901-4140-9f3c-81207d4dacc4}) (Inactive)
    Remove Temporarily 1.0.1 ([email protected]) (Inactive)
    Restartless Context Search 1.2 ([email protected]) (Inactive)
    S3.Download Statusbar 2.04 (s3download@statusbar) (Inactive)
    SaveFrom.net helper 2.37 ([email protected]) (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 ([email protected]) (Inactive)
    Test Pilot 1.2.3 ([email protected]) (Inactive)
    TinEye Reverse Image Search 1.1 ([email protected]) (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 ([email protected]) (Inactive)
    Vimperator 3.8.2 ([email protected]) (Inactive)
    Visual Bookmarks 2.10.0 ([email protected]) (Inactive)
    Vkontakte photo 0.0.0.4 ([email protected]) (Inactive)
    Vkontakte tools 0.4.3.0 ([email protected]) (Inactive)
    WOT 20131118 ({a0d7ccb3-214d-498b-b4aa-0e8fda9a7bf7}) (Inactive)
    Yet Another Context Search 1.3.1 ([email protected]) (Inactive)
    Youtube AutoPlay Stopper 1.0.7 ([email protected]) (Inactive)
    YouTube tools 0.0.0.6 ([email protected]) (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.