I'm not sure if req intended this but cats can attack the first turn they are out of barracks at least they can attack buildings, also deconstructing buildings while units in them doesn't kill 1/3 of them as destroying them can. So someone can build a tower attack and tare it down same turn with no loss. Those seem like pretty big flaws.

All units can attack when first built, they just can't move.
Cats are no different.

Can units move into a hust built (i.e, not ready) building? I thought Req prevented this...

ah i miss worded, they can he built the turn before then moved in and tore down after attacking. Still its something that is a bit of a cheap shot as is the hitting once your out of the barracks. I thought req had fixed the latter ages ago.

on top of that buildings that are under attack can be torn down, once again maby this has been going on a while but i logic that if you can't repair a building while under attack you can't tare it down either, or ot not to be able to.