1. That is why AjaxSearch, as well as all properly-designed snippets such as Wayfinder and Ditto, have customizable tpls. It makes MODx all that much more flexible.
2. resultsTpl handles the display of everything surrounding the actual list of results. That is where your "no results" text is, and where group titles and x results shown of xyz is. resultTpl is for each individual result.
3. Editing the default tpl is a bad idea. If you upgrade, that will overwrite your edited version of the default tpl. Plus if you really mess it up, you'll have a hard time getting it back to its original state. It's always better to use your own tpl; it only takes a few minutes to create a chunk, copy/paste the default tpl into the new chunk, then edit that new chunk.
4. That is correct.
5. In that default tpl, there are three conditional blocks, one for showing the description "if as.descriptionShow is 1", likewise one for showing the extract and one for showing a breadcrumb trail. Now, there is a property &breadcrumbs, which can be set to 1 or 0 in the snippet call, which if set to 0 will cause that conditional block in the tpl to not display a trail (AjaxSearch converts that property key to as.breadcrumbsShow internally). There should be corresponding properties for &description and &extract, but I haven't been able to find any reference to them. It might be interesting to leave the tpl alone and see what happens if you put &description=`0` in the snippet call.
A newbie is supposed to learn, then he, too, can become an old experienced MODx user

That's what these forums are for.