The sentences are mostly correct, just a bit awkward. Like Michael already said, "lopped off" works better together than separated in the first example. I would also say "tree branches" instead of "branches of trees". In the second example, there are 2 small problems. 1) For me personally "many" sounds too formal to go with "lopped off". I would say "a whole bunch of" or "lots of", that would match the tone better. 2) In this sentence, you need an 'of' because you're including the (direct?) object. (So you can say "When the machine went crazy, it lopped his arm off." OR "...it lopped his arm off of him." You can't say "it lopped his arm off him." ) This is according to "proper grammar"-- in casual conversation people might omit the 'of', but in writing you need it. 1. They lopped off the tree branches this morning. 2. They lopped a lot of branches off of the trees this morning.

sentence 1 would be better if you said "they lopped branches off those trees this morning" sentence 2 is ok.

2 is good. 1 is not "wrong", but it is a little clumsy. "Lop off" works as a verb. "Branches of trees" is exactly correct, but not at all something I'd expect to hear much: what else would the branches be of? In my opinion ;)