I will "bite" on this:
1. Shall I need to register 3 domains for 3 rooms in my case?
No, register one domain, and make sub domains(or applications) under it...I.E. room1.doggiesatplay.net or doggiesatplay.net/room1
2. Can I assign a user name and password to my client to filter other people to enter my webcam domain / website for privacy?
Yes, make each room accessible to specific users only! That is where the subdomains, or applications come in.
3. After a owner has taken back his dog, my order finish. Can I delete his user name of that owner and assign another user name for my new client
Yes. This would be mandatory!
4. Can I use one computer to set up 3 webcams?
This might be tricky! It depends on whether or not you can seperate each stream. You may to do some fancy high end subnetting, or supernetting! That is later.
5. Can my webcams be still working if I turn off my computer but not switch off my modem or router.
6. Can you tell we the brand ranking of webcsms
No, we have no opinion of that service.
All of this could be possible, and you could even host three Vservers on 3 VLANS if needed, but the PC would need at least 64GB of RAM!
I hope I have helped.