If you follow this guide, you should be fine: https://www.gog.com/forum/elder_scrolls_series/gog_doesnt_set_proper_last_modified_times_on_esmesp_files_which_results_in_modding_and_vanilla_i

Yes - no (according to my personal experience)

Is there, by any chance, a mod that would let you skip the starting dungeon?

Preferably, the mod should skip to the point where you choose your class and birthsign, and give you the items that you would have likely collected in the dungeon (including that one staff the goblin drops); it would also automatically progress the main quest to the point that it would have progressed had you done the dungeon normally.

(It would also be nice to have such a mod for Arena. It's not as important for Morrowind (the intro is short there) or Daggerfall (Alt+F11 immediately after starting (or loading) the game takes you to the dungeon entrance).)

No mod of this kind exists for both Oblivion and Arena, to my knowledge.
For Oblivion there exist savegames (TESNexus, I guess) which you can use to start right at the sewer exit, where you can (re-) define your character as you like. To create such a savegame myself was the first thing I did after completing the tutorial for the first time.
For Arena a savegame solution wouldn't make sense, as character generation happens before you are thrown into the starter dungeon.
Or did I miss your point?

Anyone make a hot fix for http://www.gamesas.com/missing-few-inventory-icons-t339702.html which I experienced here https://support.gog.com/hc/en-us/requests/731308?page=1&product=gog https://www.gog.com/forum/elder_scrolls_series/missing_inventory_icons_with_gog And to clarify, I'm referring to the shirt with suspenders with code 00064F77

I installed the 3 large patches to fix bugs.
Is my old save game supposed to be messed up now ? (In a not obvious way)

Im not sure installing anything is worth the trouble trying to learn buggy mod installers and search secret hints how to proceed
There are even warning about contents of those basic patches that you would need another tool to clean them:
(Wrye bash output)
...............
[Cleaning With TES4Edit Needed
The following plugins have deleted references or other issues that can and should be fixed with TES4Edit. Visit the Tome of xEdit for more information.
DLCBattlehornCastle.esp: Contains dirty edits, needs cleaning.
DLCFrostcrag.esp: Contains dirty edits, needs cleaning.
DLCHorseArmor.esp: Contains dirty edits, needs cleaning
...
................

Then I am advised to merge stuff. Nowhere is explained what that means or how to do that. lol

The only thing that was obvious is to change some loading orders. I really wonder if just leave the game as it is will make you regret it later on. (I play first time)

Maybe someone should build a basic patch installer to give people a final patch in one installer. So all they have to do after buying the final edition here is launch that one installer and be fine with it. No everyone wants to dive into the mod world.

It really depends on which and how many mods you use simultaneously. Generally, the more mods are active in parallel, the higher the risk of conflicts between mods, and the higher the need to take care of how you load them. There is also the issue that savegames created with one set of mods active may not work when a different set of mods is active.

For my own purposes, I always keep an unmodded Oblivion installation and one installation per mod configuration, together with a savegame manager helping me to keep the savegames for the different installations separate. Some of my installations use tools like OBSE, Wrye Bash or LOOT, others don't. If I want to try a new mod configuration, I use a new Oblivion installation, which avoids screwing up existing ones. If there is a problem, I sometimes use a mod conflict analyser, too,

If you simply want to play Oblivion and/or Shivering Isles, the base game plus the unofficial patch(es) are fully sufficient.