Primary function regarding email check is getmxrr or gethostbynamel (if getmxrr fails), which some hosts disable and some mailservers disable and some email servers don't respond to. Typically it only shows "error" when the host or mailserver has this function disabled. The username check only fails if the database query fails.

Please install recently released CB 1.7 stable and confirm if issue still present.

I think this maybe more of a URL issue as in the ajax call is not calling the correct URL. Maybe due to SEF or an htaccess rewrite, unsure; if any are present then please disable them and re-test.

GoDaddy in 4GH doesn't support those functions. Where those introduced in 1.7 as 1.4 worked fine?

No, they have always been there.

How "bad" is it if those don't check (I don't mind the email) but why would the username error out too?

It's not bad at all. It's purely for your convenience and has no relevance to CB functioning or not.

I haven't gone up to 1.7 stable yet.

Please upgrade to CB 1.7 stable as soon as possible.

The ajax checking is purely JS side. The PHP end won't allow duplicate emails or usernames. The issue is more or less cosmetic. The checking wasn't changed with 1.7 except for a bug that caused massive delay in the checking (perhaps why it is erroring? is because it can't complete?). Not all hosts will support the functions needed. At any rate have created a bug ticket for further review of the issue.

forge.joomlapolis.com/issues/2700