1) Is it possible to change parts of a page only by clicking on a button?
Generally speaking, yes ;-)! You could for example call a custom action plugin which modifies a page if a certain action gets triggered by your button.
2) Where should I go? Using something existing (command plugin) or create something from the scratch?
OTOH I don`t know a plugin which does something similar. AFAIK the poll plugin uses a action plugin to store poll votes in a separate meta data file. Maybe that gives you some ideas.
2) Can I recycle that code later to use Ajax functionality?
Yes, if the plugin which shows the output relies on page metadata rather than on the page content itself. The poll plugin does something like that but without AJAX (it reloads the page after someone voted). If you want to rely on AJAX you don`t need an action plugin, a syntax plugin would be enough. The "button" of your plugin would trigger an AJAX action which would update a custom meta data file and then reload/re-render a certain part of your page. However, doing what you want without AJAX is a lot easier to implement.