I hope you have a good time Glacier and happy to pass along what little advice I could. I've been to Ireland and Austria also and I think Berlin might be the place I'd go back to first. I had three days there and it wasn't nearly long enough. Where is your hotel going to be at and what airport are you flying into?
I flew into Tegel I believe, which from a quick Google check, appears to be permanently closed now. But coming from California and having a layover in Amsterdam, the arrival to Berlin could not have been any easier. I cleared customs in Holland since that's where I entered the EU so Berlin was a charter bus right to our baggage carousel, walk a short ways and go out some like side door and then bam, there's all the cabs.
One story that's only amusing to myself and I'll try and keep it short. So I'm planning to go on the Insider Tour that I mentioned on my first day there. I'm thinking I won't have much jet lag and as long as I don't fall asleep and sleep the day away (like I did when I arrived to Dublin, Ireland), I should be ready to go. I'll shower and then I know where the meeting point for the tour is in West Berlin.
I shower up, get dressed and I'm out the door. I come out of the hotel and I take a left and I'm looking for the train station. I'm walking and I'm in kind of a residential area. There's some apartment buildings and some restaurants and I'm enjoying the walk but I don't ever find the train station. So I turn around and I walk back to the hotel and up ahead like 50 yards, there's the S-Bahn station

Had I just come out of the hotel and hung a right, I would have found it very quickly haha
Anyway, in regards to Checkpoint Charlie, I think I might have gone there or near it but I can't remember to be honest. I believe there are still parts of the wall up that you can visit.
Two sites that I would suggest checking out are Fodors.com and TripAdvisor.com. I haven't been to either site in a couple years probably but both have/had really good forums where you can ask questions and get a lot of info and advice.
And in regards to the gate, I want to say that I'm positive that you'll be able to go right through and there won't be any kind of fee. I still don't know why I didn't walk through myself. I had met a Kiwi on the tour (we might have been the only two that spoke English

) and when the tour ended at the gate, we just went back and caught the S-Bahn to Alexanderplatz and got something to eat. I guess I'll just have to go back someday and walk through.
And I'm not sure if you will have the free time to do it, and even if you did, it sounds like you have other interests, but here is the link for Insider Tours. I'd definitely do another one of their tours in a heartbeat.
https://insidertour.com/