How to fix this backsplash tiling job?
For the tile spacing issue, I don't think there is much you can do at this point short of ripping it all out and starting over.

For the tiles not set in plane correctly, I think you could carefully remove those tiles (just the individual tile), scrape out the hardened thinset behind them, and then re-apply with new thinset.

For the thinset squeezing out, perhaps you could grind that out with a oscillating tool with a grout removal attachment.