To pair and connect the HC05 only needs 5V power and ground. Disconnect everything except from 5V and ground from the HC05 and try it. DO NOT put the HC05 into AT mode and try to connect.
If the HC05 still will not pair with everything but 5V and ground disconnected and in communication mode, there is a fair chance that it is no good. There have been instances of bad modules lately.
If you did connected any of the HC05 inputs (RX, EN) directly to 5V there is a chance that it is damaged. The inputs are not 5V tolerant (despite what you may see on the net) and can be damaged by 5V.
I've seen people on internet (mostly youtube tutorials) those connect to 5V directly so I did that, I managed to pair and send/receive data using python for testing. But I tried my mouse connected to usb host shield and mouse inputs were incredibly laggy any ideas?