Really any kind of sites?

My main cms - Joomla. I’m using it more than 10 years.
I found Grav and I want to try it.
Is it really to build any kind of sites?
Is it possible to use extra fields and filtering in frontend, like E-Commerce sites?
Is it possible to create forum with Grav?
Is it possible to create files downloads site with counter for each file?
I just can’t imagine how it possible without database.

Yes to all of the above, though not out of the box. Fields, filtering, sorting, and handling many items for an eCommerce-site isn’t all that difficult - you’ll accomplish all that with Twig or JavaScript. You’ll need a payment-operator though, but there are at least two plugins which can provide this.

There is not an integrated forum-plugin, but there are OAuth- and LDAP-plugins which you can use to authenticate with both Grav and most modern forum-solutions, and they can run alongside each other without issue.

You can create download-counters, but there are no built-in solutions that I’m aware of, so you’ll likely have to code it yourself.

The traditional database is just a storage-layer working with the CMS. There’s nothing in a storage-layer that cannot be accomplished using files, key-value stores, or any other type of storage as long as it can receive and give back data that the CMS can use.

For heavy-use sites, or intensive operations like large eCommerce sites or popular forums, I’d want to go beyond flat files regardless - and also invest in more extensive caching, load-balancing, routing and asset-handling. Grav can still be the presentation-layer, like pretty much any CMS, it will just require less resources than most.

Is the current forum builded with Grav or it’s 3-rd party engine?

It’s based on, rather sofisticate and complex discussion software.