[RPG] Guidance for applying Advantage/Disadvantage to attack rolls vs. contests and saves vs. checks

Yes, A DM could impose disadvantage and grant advantage on the same contest.

There is precedent for it. The Cloak of Elvenkind grants both advantage and disadvantage to a Wisdom (Perception) vs Dexterity (Stealth) contest. From the SRD:

While you wear this cloak with its hood up, Wisdom (Perception) checks made to see you have disadvantage, and you have advantage on Dexterity (Stealth) checks made to hide, as the cloak's color shifts to camouflage you. Pulling the hood up or down requires an action.

It is therefore possible to imagine other situations in which a Dungeonmaster could grant similar concurrent bonuses/penalties. Note that items which do this are very few. A ruling such as this based on a current situation should be just as rare. Advantage vs Disadvantage contests are pretty much a foregone conclusion, and almost beg the question "why bother to roll?"

Yes, advantage would negate disadvantage for purposes of Sneak Attack.

According to the rules on advantage and disadvantage (PHB 173, emphasis mine):

If circumstances cause a roll to have both advantage and disadvantage, you are considered to have neither of them, and you roll one d20. This is true even if multiple circumstances impose disadvantage and only one grants advantage or vice versa. In such a situation, you have neither advantage nor disadvantage.

So, for example, if you (the rogue) have both advantage and disadvantage on an attack roll against an enemy while your ally is within 5 feet of the target, you could make a Sneak Attack because you are treated as if you don't have disadvantage when making the roll. This complies with the requirements for Sneak Attack (PHB 96, emphasis mine):

Once per turn, you can deal [extra] damage to one creature you hit with an attack if you have advantage on the attack roll.... You don't need advantage on the attack roll if another enemy of the target is within 5 feet of it, that enemy isn't incapacitated, and you don't have disadvantage on the attack roll.

However, if the ally were absent in that situation, you could not make a Sneak Attack because you are treated as if you don't have advantage when making the roll.

The same rationale would apply for features other than Sneak Attack that depend on either advantage or disadvantage on any attack roll, ability check, or saving throw. When any such roll is made with both advantage and disadvantage, the roll is made as if it actually has neither advantage nor disadvantage. In this case, no feature which requires advantage or disadvantage would be activated, and no feature which forbids advantage or disadvantage would be precluded.