Regarding Advantage and Disadvantage, in the games I run and play this mechanic is typically not over or under used. I think we have found the sweet spot, but this is likely due to our group's particular play style. Additionally, we don't allow flanking to provide advantage, which really helps. One thing I've started incorporating is allowing a 1d4 bonus / imposing a 1d4 penalty. This can be used in conjunction with advantage/disadvantage and creates a larger spectrum in terms of interacting conditions that might affect the outcome of an action.
2. CRITICAL HITS (and Ability Checks/ Saves)
2A. On a linear d20, if you need a 20 to hit a creature, it should NOT also be a critical hit; it should be a normal hit.
Personally, if an encounter includes a creature that one or more players can only hit on a natural 20, I would consider that a poorly designed encounter. It's not fun for players to miss more often than they hit and generally just leads to frustration. Of course there could be circumstances where this might happen, but that should probably be an exceptional situation that
might happen once in a campaign. For me, I would rather a DM tell me it's impossible to hit the creature than to only allow it to be hit on a natural 20. Of course, this is my perspective and may not be right for your personal play style or game table.
In the games that I run, I handle critical hits a bit differently. I do require confirming a critical hit, like they did in 3/3.5. However, a crit deals maximum damage, and a confirmed crit does double max damage and the player rolls on a d100 table for an additional effect. I do the same thing with fumbles, but it is just a miss unless it's a confirmed fumble. If it's a confirmed fumble, its a miss plus they roll on a d100 for an additional fumble effect. I feel this makes crits incredibly deadly, makes them more interesting than just dealing pure damage, and it reduces complexity by having crits deal either max or double max damage (which also ensures a person doesn't feel salty when they roll low on their critical hit). I also apply critical success and failure to saving throws and ability checks.
Whether or not crit fumbles or saves are necessary depends on your play style and game table. I don't think they are necessary. However, I do think they add an additional element of excitement to a game. It can also make encounters a bit more swingy (which I personally like, but others may not).