Yes, with no session limit, no duration cap and no “commercial use detected” warning. There is no paid edition.

Can somebody connect to my machine without me noticing?

No. An incoming connection has to be accepted on the machine being controlled, and the session ends when that machine's user ends it.

Do I need to open ports on the router?

No. Both sides connect outward, and the server introduces them. It works behind NAT, behind a mobile connection and behind a corporate firewall that only allows outbound traffic.

Does the video pass through the server?

Only when the two machines cannot reach each other directly. When a direct path exists, the media takes it and the server sees only the negotiation.

Can I run the server myself?

Yes — that is the point of publishing it. Your own server means the account list and the connection metadata stay with you.

Does an account create any obligation?

The account exists so that the address book has somewhere to live. It is free, it is not tied to a payment method, and a one-off session needs only the nine-digit number.