K2 frontend editing uses a predefined format which is similar to the item form in the backend. In the K2 component settings (under the tab "Frontend Editing") you can control what is shown in the edit form. However, if you wish to fully customize the form, you can just override it inside your template.

- Copy the file /components/com_k2/templates/default/itemform.php inside /templates/YOUR_TEMPLATE/html/com_k2/default/itemform.php.
- That file actually includes another one (from the K2 backend). We do this to only maintain a single edit form. So open the file /administrator/components/com_k2/views/item/tmpl/default.php and copy all of its contents and paste them (to replace) the contents of /templates/YOUR_TEMPLATE/html/com_k2/default/itemform.php.
- Then open the file /templates/YOUR_TEMPLATE/html/com_k2/default/itemform.php and start editing the form as you like.

Make sure frontend editing is enabled in K2's component settings and that the K2 User (not K2 Users) module is enabled somewhere in your site so that you can see available edit actions. Additionally, you need to have at least one user configured for frontend editing. These rules are defined inside K2 User Groups. In there you can set which K2 ACL is available to a given K2 Group.

If you configured the above correctly, you should be able to see add/edit links for content in K2 views and on the K2 User module.

As for showing a different form per K2 category, I would recommend you create sub-overrides for itemform.php like this:
/templates/YOUR_TEMPLATE/html/com_k2/default/itemform.php
/templates/YOUR_TEMPLATE/html/com_k2/for_category_1/itemform.php
/templates/YOUR_TEMPLATE/html/com_k2/for_category_2/itemform.php
...and so on

Then create menu items for the K2 Edit Form and make sure you select the relevant subtemplate.

That's the easy way.

The slightly harder way is to hardcode K2 category ID checks inside the itemform.php override.