Misteaks? I dont kno Anything about any Mistaks. Because Im Perfec.

I have a question about author sloppiness. I have recently gotten my hands on quite a few uncorrected proofs and I've noticed that some of them have very few errors - maybe as few as one or two, but others have many. Not just typos, but lots of formatting errors, and even continuity errors. I realize that they are uncorrected, but it brings me to ask this question. How much does an author's sloppiness affect the ARC and the final book?

This depends on the editor, copyeditor, and proofreader. Sometimes a mess of a manuscript inspires the publisher's team to go through it with a fine-tooth comb, and other times it inspires them to think, "the author clearly didn't give a crap; why should I?"
I'm not defending that thinking, fyi.

For example, someone in my critique group is notoriously sloppy with typos and continuity errors, and when her first book came out, there were more errors in it than I've seen in any other finished book. I'm just wondering how much a writer can actually rely on editors and copy editors? My agent has recently sold my book and I'm on the first round of edits, so I'm a ways off from this, and I'm generally quite meticulous, but is there anything I should do besides try my best to get things right and also have someone else read it who is detail oriented?

That is the most you can do, and that much is deeply appreciated by your editor, assuming you have a decent one.

Or are some of these errors coming later, AFTER the writer has handed over the project?

Yes, of course some of them are.
No, don't freak out. This happens to pretty much every book as it goes through the process. That's because it's a more complicated process than most people realize, and between communicating edits among a team of 4-5 people and transferring text from a word doc to whatever the designers are using, mistakes can and will happen. That's why there are multiple rounds of galleys and proofs.