The plum tree should be pruned every year - you can find help on proper pruning online or at the local nursery.
You can cut branches out anytime of year. But, plum trees are not large trees so cutting lower branches may lead to less fruit and the need for a ladder to pick it. When you look at the point the branch is attached to the trunk, there is what looks like a little collar. Cut to the branch side of the collar.
No, the first tree is not a plum tree.