Posting part of novel for online critique - will I still get offers?

I don't have a definitive answer for you, but yes, if you're looking for representation by an agent, I'd suggest caution at least.

Share Your Work here on AW is password-protected, and is therefore not googleable (as I understand it). I'd be comfortable posting there. I don't know about querytracker or scribophile, but users there should be able to tell you.

Anything posted in the SYW section of AW is fine. It's password-protected and doesn't come up on Google searches, so it's not considered published. I don't know about other sites, but check their terms or ask their mods to see if they have some protection in place.

Posting a single chapter or two in public is unlikely to cause copyright issues or reduce your chance of getting an agent or published later. But do keep in mind that anything you post in public is likely to be able to be churned up from the dark underbelly of the Internet in later searches. Think before you post: do I want this representing me forever? If it's not in great shape, stick to protected critique sites.

Good point, I haven't thought of that.

However, the SYW (Share Your Work) forum here is truly excellent. The writers who give advice have good suggestions. You might not always agree with the advice, but there's lots to think about.

Perhaps also send the opening to a few more Beta readers too? There are paid services who do critique and assist - I can't afford them, but if you can, it might be worth it if you're concerned about copyright issues. There's also something that the MSWL Academy have started: http://www.manuscriptwishlist.com/2017/02/want-to-spend-10-minutes-with-an-agent-now-you-can/
It's a paid service, though.

Good luck with your book.

There's an FAQ for that.

Am I using my first time publication rights if I post for crit in Share Your Work?

There are actually several kinds of rights, including several kinds of first rights.

https://www.thebalance.com/north-american-publications-rights-1360551

It's rare for an author to publish all of their work online. In fact, it's almost standard procedure to post the first few chapters of a book online. These excerpts act as an ad for the completed work. After you've sold the book to a publisher, Amazon and B&N do this for you, and as long as you don't publish any more than they do on your or someone else's web site they won't come after you. (I made that mistake & got a warning from Amz. After I cut back my excerpt to match theirs they approved my practice.)

Even if you use up your first right by posting the entirety of your work it on your web site, you still have all other rights. John Scalzi did this with his first two novels. They were discovered by Tor Books, who bought the two books. They launched his career.

https://en.wikipedia.org/wiki/John_Scalzi#Fiction

Drafts of parts of what eventually became Fifty Shades etc. were first published as fan fiction. Reworked into complete works they were published by a small publisher. They could not meet all the demand for the work. It was resold to a trade publisher, which could meet the demand. Details are at the link below.

https://en.wikipedia.org/wiki/Fifty_Shades_of_Grey#Background_and_publication

Some agents might have problems with either of the practices described above. But I suspect most would be willing to put up with any pain if they felt that a book would meet a wider and higher priced market if they worked through any problems. Wider markets include translation rights and film adaptation rights.