Our men would be Οι άνδρες μας, I believe. Οι δικοί μας άνδρες gives emphasis on the fact that the men are ours.
"οι άντρες μας" means either "our men" or "our husbands" depending on the context. As for so many new words, first of all, each lesson only has one new word and secondly, you'll get lots of chances to practice so you'll get used to them.
I think one of the reasons you're finding the words a problem is that you have been completing all the skills before going on to the next one. What those of us who have done many lessons on Duo have discovered is that if you do only part of a skill then go on to another one and go back to the first one you'll find it easier to learn the words.
So, for example, I see you have finished five skills and are about half the "Possessive Skill". Well, you should leave Possessive and go on to "Clothes". Do a little of "Clothes" and go back to "Possessive". You'll find it much more relaxing and you'll remember the words better. Give it a try.