I am unable to add new tags from the front end.

You have to hit comma or enter key to add a tag.

Getting 500 errors when adding new question. Why?

As I said before AnsPress is not tested with multisite, and it may not work with multisite. But I would like to investigate the issue. Will you confirm that this error is not caused by other plugin, disable all plugin and then try.

Side bar for Category page?

You can use Ap Category position

Does Anspress support multisite?

For those who have tested multisite, did you have to change anything with permalinks or the URL structure? For example, ‘Ask’ takes me to “…/questions/ask/” on the main site, but on my other sites it takes me to “…/2015/09/29/hello-world/ask/”. Seems to be creating a new blog post or something along those lines. Any suggestions on how to fix this?

Can I list out all the category make it like a forum system?

Yes you can set logo for every category.

How to modify login widget?

You can edit login-signup.php in AnsPress theme.