Combining a conference with a personal vacation

Yes, it is typical to take personal time before or after a conference. Funding from your university would not usually mind.

Conference funds may be a little less flexible, so you should certainly check with them. With evidence in hand it will be cheaper for them, I find it unlikely they would decline.

This is generally acceptable, but might be forbidden in a few places. It isn't an ethical concern, however. As long as you are clear in how you use the funds, properly separating expenses and as long as you fulfill the obligations of conference attendance you should be fine (ethically).

But if there is someone appropriate that you could check with, it might be wise to avoid questions later.

It would, of course, be unethical (and possibly illegal) to use the funds and not attend the conference as expected.