Does the Abyss exist independently of difficulty modes, or are there 5 different versions of it?
No, it doesn't matter which difficulty you start the Abyss you will always start at floor 1.
Floor 1-5 correlates Way of the Samurai.
Floor 6-10 correlates to Way of the Strong.
Floor 11-20 correlates to Way of the Demon.
Floor 20-30 correlates to Way of the Wise.
Floor 31+ is Way of the Nioh difficulty and you will have to complete the Queens Eyes on Way of the Nioh to unlock this and beyond.
I am currently at floor 53 and liking Abyss mode. Quick, repeatable content. You will be afflicted with 4 debuffs. There are 4 boats behind you going to a mini level that you can complete to get rid of a specific debuff. The colour of the debuff and boat flag corresponds to which one you will get rid of.
Most of the time you can easily just ignore the debuffs and rush the boss especially at lower levels. However, there are some nasty debuffs like being permanently afflicted by sloth or sludge (can't use items). In that case, I would suggest completing a plane (boat level).
You will gain 'defiled' gear which do not have stats and are unusable. You will also gain an item called 'droplet of decay' from bosses and you can use that to defile your divine items. When you've had enough and leave the Abyss your defiled gear has a random chance to turn into ethereal versions of it.
Be warned, if you die your guardian spirit will not only hold your amrita but also your defiled items, so always try to regain it quickly. There is also an item called bell of invocation that you can use to regain your GS, amrita and defiled items too.
When you leave you can also start of at the highest floor you have completed.