Can you re-wire an EU house with UK sockets?
There's no reason why not so long as you install everything to the local regulations and ensure proper fusing, etc. Ring Mains are a uniquely British circuit so you'd be looking at radial circuits so you would need to check the fusing (circuit breakers) are an appropriate rating for your sockets. Also bear in mind that if you have no local sockets, buying a new appliance with a Bulgarian plug on it is going to be frustrating.

You can fit whatever sockets you like to an installation so long as you do so safely in accordance with the local electrical regulations. That doesn't mean you can just replace all the sockets without checking how they are connected and protected. If you're not competent to follow the appropriate regulations, hire an electrician to do it.

Done properly with decent quality sockets (e.g. MK) this is going to be a lot safer than using those shonky travel adaptors etc.