[verb て-form] + も構わない [i-adj 〜い → 〜くて] + も構わない [na-adj] + でも構わない [noun] + でも構わない [verb ない-form 〜ない → 〜なくて] + も構わない
ても構わない says something is acceptable, or that the speaker doesn't mind it: "it's all right if ~", "I don't mind if ~". A verb takes も構わない after its て-form, an い-adjective becomes くても構わない, and a noun or な-adjective takes でも構わない.
Both give permission. ても構わない is a step more formal, and it says the speaker has no objection rather than that the thing is good. That makes it the choice at work and in service settings, and for accepting a drawback.
構いませんか asks "would it be all right if ~?". It is literally "you don't mind?", so the reply that grants it agrees and starts with はい or ええ, where English "Do you mind?" gets "no".
With a question word, or about something that could go badly, ても構わない says the outcome makes no difference to the speaker.
These go beyond giving permission.
The negative becomes なくても構わない: "it's all right not to ~", close to なくてもいい.
A noun or な-adjective takes でなくても構わない.
About your own action, ても構わない offers to do it.