Do the different Kart pieces affect gameplay

Short answer: No

The decision to remove wheelies from Mario Kart 8 was an intentional one. The reason stems from the fact that, due to poor design decisions in Mario Kart Wii, the bike became an over-powered vehicle. This is a statement attested to in general by players and developers alike.

The main factors that over-powered bikes were:

• Overall Speed - Bikes generally had higher speeds than karts. The two fastest combinations in the game both require bikes.
• Wheelies - Gave them an additional speed advantage that karts did not have.
• Inner drifting - Allowed for sharper curves that karts could never perform.

The developers of Mario Kart Wii attempted to balance this situation by limiting how bikes boosted via drifts: granting only blue boosts, instead of the more powerful orange. This however had little impact.

Enter Mario Kart 8

The developers have made every attempt to balance bikes, while still retaining their spirit. The first and most obvious change in gameplay mechanics is the removal of wheelies as you noted. This makes them more on par with karts.

Other mechanical changes to bikes in Mario Kart 8 include:

• Increasing the drift angle to be less sharp.
• Giving them access to orange speed boosts, just like karts.

All of that said, as Matt has stated, when boosting via mushrooms if a player is using a bike in Mario Kart 8, the character will perform a wheelie. This is simple eye-candy and has no known effect on gameplay.

Yes.

The number of coins added to your total is the number of coins that each local player has at the end of each race. Regardless of how you collected a coin, if you have it at the end of the race, it counts.

For more details see the Mario Kart 8 Total Coins Collecting Logic question.