Add own profile tab with own content

Yes you can, simply use this function in init hook: ap_register_user_page($page_slug, $page_title, $func, $show_in_menu, $public);

Category shortcode not working?

I think this is due to other search filters. Question embed and question as comment will introduce in 2.4

Logo Category in Home Side Bar

Check this image:

Notification and profile

I told you to read docs. Its in appearance->menu->anspress->user notification. Please avoid duplicate questions.

Which PHP files do I need to modify in order to add a new field in the question form?

https://gist.github.com/rahularyan/0b37680c1a24c1e78baf#file-gistfile1-php

the title question is double

Edit AnsPress base page and set page template as AnsPress page.

category post excerpts

Its look like you have modified content.php?

Logo Category in Home Side Bar

You are using image in category, so its usual. If you want small icons you must use font icons. You can check available fonts in askbug here: Font icon

Viewing a QUESTION loggs me off

AnsPress is not responsible, check other plugins.