You didn't say how far down the tube the cut was, nor how deep it was, but the butting goes a fair way down, usually 125mm or 150mm, and then there is a tapered section before you get to the 0.6mm thick part. I'm no expert, but if the cut wasn't too deep and if it was in the 1.2mm thick section, then hopefully you should be ok?
And as for the dent, well again you've seen it and we haven't, but I can't see it from the outside, so hopefully the previous comments apply and it shouldn't ever cause a problem. How did he manage that though? I thought I was ham-fisted, but I don't think I could manage that if I tried!
I guess any potential for a problem would be minimised by using a nice long post inserted well down the tube?
It would be a shame to have to have the seat tube replaced, but hopefully the need will never arise.
The cuts run the entire length of the butt and onto the thinner tubing below. Difficult to measure how much material has been removed along the length of said cuts. At some points it looks deep(ish) at others it looks to be little more than a scratch. Think the previous owner stuck something down the tube to try and pull the stuck post out from below, hence the dents. They aren't of concern having verified they aren't cracks.
Running a decent direct fit (ie not shimmed) post and seat collar is the way forward.
As for thompson (sic) being last year I disagree. It's great stuff and reasonable value - they've taken the mantle of what syncros used to stand for in my book.