Yes, they sound similar, but not identical. The grammar can usually be of help, but ultimately you're going to have to learn to tell them apart by sound as well.
"mannen" is pronounced without an 'e', but the last 'n' gets drawn out and stressed twice, making it a two syllable word: "mann-n". It's very subtle at first, but as you progress with the course I think you'll find it easier to make a distinction.
The way "mannen" is pronounced in this sentence sounds particularly vague to me, making it a poor example, so if you can't hear it here then just try again with another sentence. :)
The 'e' is normally dropped, yes. The last 'n' is still there if you listen closely, but since the syllable just before it ends in an 'n' as well it can be a little tricky to hear.
Listen for a long, drawn out 'n' that is stressed twice. The audio for this sentence pronounces "mannen" in a way that's particularly vague, so you might want to try with another one first. :)