'On a cut' implies that the period of caloric restriction has a planned end-point (ie, either a specific weight/body composition, or a specific date). Your goal, I assume, is to maintain as much of your hard-earned muscle mass as possible during the cut. Others may disagree, but I think the key to convincing the body to hold on to muscle despite a caloric deficit is to regularly apply extreme tension to those muscles; ie, to lift heavy weights.
So to answer your question, at no point during a cut would I lower the weight and increase volume (with the exception of a planned deload week if your program calls for such, after which you'd go right back to lifting heavy again).
As for how many heavy sets you can/should do, that's a function of your training history and workload capacity, as well as the severity of the caloric deficit you're running (you can tolerate a lot more volume on a 500 cal/d deficit than a 1000 cal/d deficit), and for how long you've been in a deficit. There are no hard & fast rules for determining this, I'm afraid.
As for getting stronger--once you've been in a deficit deep and long enough, doing so inevitably becomes impossible. (I'm assuming you're 'un-enhanced.') Once you find strength gains are no longer coming, just try and stay as strong as possible for as long as possible.
