Can I put AdSense or other ads on my site?

Of course. You can put whatever ads you want, just paste your code in the template.
The template files can be found in application/views/frontend/themes/[your-theme]

What hosting should I choose?

You should be fine with almost any Linux hosting.
We could recommend though as a good hosting provider. You could use our affiliate coupon code CLASSISCRIPT for 25% discount.

Details about what the hosting requirements are can be found in the installation page.

Can I modify the source code?

You can modify the layout (templates, CSS, JS files, etc.) and some parts of the backend functionality which is unencrypted.
However the main controller file (application/controllers/main.php) is encrypted. If you need to make some change and can't do it without changing this file, let us know and we'll see if this could be made as an update - in this case we'll do it for free.