Cardio isn't needed at all during a cut. The only thing that is needed is a calorie deficit. There are two ways to do this:
1. Preferably go for the foods that are highly satisfying and low in calories. Examples are fruits (less sweet ones), vegetables, and lean meats.
2. Eat less and/or less often.
Gaining strength while cutting will be dependent on where you're at. I have done it before. But like I said, it really does depend. If you're out of shape then no problem. But if you've been working your but off and are the strongest you've been in your life, I wish you lots of luck, because you're going to need it!
Either way, you should at least try. You can't do it if you don't try.