[RPG] Do you regain consciousness immediately after rolling a 20 on a death saving throw

Yes

From the Rules Compendium, p260 (emphasis mine):

Death Saving Throw: When an adventurer is dying, make a saving throw at the end of that character's turn each round. A death saving throw works just like any other saving throw, but the result determines how close the adventurer is to death.

If you get a bonus to all saving throws, that bonus does apply to death saves.

The easiest way to gain advantage on a death saving throw is to use inspiration.

If you have inspiration, you can expend it when you make an attack roll, saving throw, or ability check. Spending your inspiration gives you advantage on that roll.

Alternatively, you could be a Wild Magic Sorcerer, and use Tides of Chaos. Well, probably.

Starting at 1st level, you can manipulate the forces of chance and chaos to gain advantage on one attack roll, ability check, or saving throw.

The 3rd- level spell Beacon of Hope explicitly grants advantage on death saving throws for its duration.

Fighters can use Indomitable to reroll a failed saving throw - not precisely the same as advantage, but pretty close.

Beginning at 9th level, you can reroll a saving throw that you fail.

There are almost certainly more options that I've missed - getting advantage on things is a pretty fundamental part of 5e.