Certainly - ignore their description. You're virtually correct in that all three are examples of early beaded border pattern F763 but lots 2833 and 2972 are Gouby A+b where the R & E of REG touch at the base and lot 2834 is Gouby A1+b where R & E do NOT touch at the bottom. All 3 are rare and should fetch broadly similar prices.
I don't know. It does look genuine to me. No sign of anything obviously fake.
With that said, you may find http://www.coinauthentication.co.uk/ useful.
The colour looks too uniform for my liking. Even with a full lustre coin you see some variation in the toning. It ought to be 175 years old, after all.
Having said that, it is the commonest date. Question is, are the marks the same on all coins?
I've never heard of any 1841 halfpenny fakes, nor indeed any non contemporary copper or bronze fakes (tooled to look like a 1933 penny, excepted). Although it wouldn't surprise me if in due course some appeared. Maybe others know differently.
What would concern me here is the fact that you got a different coin from the one shown in the buy it now pic. I'd take that up with the seller - do you have a link to the sale and a pic of the reverse of the coin you have?
Thanks.