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Sounds like Rocket: Robot on Wheels.

Might be King Rabbit.


https://www.youtube.com/watch?v=ANomRsnvBK4

Or maybe Furdemption.


https://www.youtube.com/watch?v=2tCAC9U2Exk

Box pushing game could be Sokoban or a clone of that.

Sounds like Fire Ant.

There was a fairy, sometimes two flying behind the main character in certain points of the game.

https://youtu.be/cEClHhd9ieU?t=550

Not sure, but reminded me of Legend of Hero Tonma, or Magician Lord.

Looks like Super Dimensional Fortress Macross.


https://www.youtube.com/watch?v=V0Kw-UVH_1g

Well with three guesses, one had to be right. :lol

Not sure, only games I can think of are DJ Boy, Skitchin' or Greendog: The Beached Surfer Dude!

Only one I can think of is International Superstar Soccer 64. It is available from EP's main site.