Should you overcall with a (slightly) weaker hand if it is more "disruptive?"
I like to think that 1-level overcalls have three purposes, at least two of which should be fulfilled to justify the overcall:


Showing values
Directing the defense (ie showing a good suit you want partner to lead)
Preempting the opponents


So eg I'm happier to overcall 1S over 1C with a weaker hand than over 1H.

2-level overcalls are trickier. It is far easier for opponents to penalize you at the 2-level, and it's more difficult for your side to make the game-partscore decision when you have the bulk of the high cards because there's less bidding room.

One rule I've heard is that a minimum opener with 5332 shape is insufficient for a 2-over-1 overcall, but many ten-counts with a good 6-card suit are sufficient. And it does seem reasonable to stretch a little to bid 2C over 1D, but not too far. As a friend of mine likes to say, 2-level overcalls are serious business.