Does the ability Contrary have any impact when using Psych Up

The Power of an attack directly correlates to how much damage you deal, because its presence in the damage formula is entirely multiplicative. So, an attack with 120 power will do twice as much damage as one that has 60 power. The same goes with your corresponding attack stat: a Pokemon with 100 Attack will do twice as much damage as one with 50 Attack when using the same attack on the same target.

STAB has operated consistently since Gen 1 so this applies to all of the games. It increases the Power of an attack by 50% if it matches one of your types. 60 becomes 90, for instance.

Technician increases the Base Power of an attack by 50% if its Power is 60 or less. So the most you get is boosting an attack at 60 power to 90. This occurs before STAB is calculated, so your Power would be 90 for STAB, boosting it to 135.

Yes, the moves that can hit you have changed over time:

• In Generation I, Bide, Swift and Transform were all able to connect (however, starting with Pokémon Stadium, Bide lost this ability again).
• Generation II replaces this list with Gust, Thunder, Twister and Whirlwind. Additionally, Gust and Twister do double damage against a flying opponent, and any move can hit if Lock-On or Mind Reader was used the previous turn.
• Generation III adds Sky Uppercut to the list of unavoidable moves.
• Generation IV lets all Pokémon with the No Guard ability hit a flying Pokémon with any move.

Presumably, the Generation I list was actually a bug which caused the "semi-invulnerable" check to get skipped (with Bide getting fixed for Stadium); the list used for Generation II and on were selected because it makes sense that these moves can still hit while the opponent is flying.