When doing mayhem with stomp only, don't sprint. Rather jump immediately after stomping then airdash forward a bit and stomp again. Easy points if you do this in a straightline down crowded sidewalks (rubbish bins, postal boxes, etc...). And in general Mayhem strategy, I usually completely ignore targets marked in red, rather I go after anything clustered together. Fences are great, as well as the things mentioned above. Its more important to keep combo going then to get individual higher value objects.
For the power-related kill challenges, an easy way to farm alien kills especially after beating the game is to load up the hard fightclub in the middle of the city, get to round 2 and don't immediately try to take the hill. You will get swarmed by an unending supply of enemies to kill. I didn't realize this until doing the 1000 power kill achievement and I've been getting around 100 kills easy each time. Just make sure to leave enough time to finish the next two rounds.
Last and probably obvious but for some of the really hard ones like Speed Rift, if you just can't Gold them, lower the difficulty. You will still get credit regardless if you play on easy/normal/hard and get the achievement either way. I went from barely getting bronze on some activities on normal to easily golding them on easy on first try.