With stone fruit trees you are best to prune them in late summer. Pruning over winter (as you would with other fruit trees) can make them susceptible to disease infecting via the cut branches.
I don’t have a cherry tree, but the rule of thumb is to open a tree out with main branches spread evenly over the head of it. You then prune side shoots back to make spurs from which fruit should grow or cut them right out if there are too many. You can prune shoots from these side shoots into even shorter spurs the following year.
Maybe take off some of the very congested new growth so you can see the tree’s centre and make a plan from there. You can always do the pruning in several goes and have a think in between as to whether to take out main branches or not.
As for the disappearing fruit, I would assume bird damage. You have to be very thorough with your defences to keep them off. One of my plot neighbours has a cherry planted in a fruit cage and I have had to let blackbirds out of there several times. They will squeeze through the smallest gaps if there are cherries on offer