I have a very nice Cayin tube amp CS 55A with EL34 tubes who uses a 8 pin socket. But I am wondering if I could use this 8-pin to 4-pin adapter (or other adapters) and use a 300B tube?
One thing is the socket, another is if the amp can use the 300B tube with no issues.
I would not do this, 300b are very picky and the amp isn’t designed for it. It won’t work because the amp runs in ultra linear (pentode push pull), and 300b can only operate in triode, it might be pretty ugly lol. Also there are voltage and bias differences as well. Just stick to el34 and kt88 imo (my preference leans towards kt88). Also not using the expensive tubes like 300b is kinda a bonus lol, greatly reduces the cost of putting good tubes in there
I am not surprised over your reply, I kinda expected that, but hey when you get an idea, better find out if it is feasible 
You’d need to know what is inside that adapter and your amp to even try it safely.
The 300b is a 5v heater, and the EL34 is a standard 6.3v.
I’d want to see how they are converting the heater power to the directly heated triode. 300b’s have no separate heater pins, you have to provide a voltage differential over the 2 cathode pins centered around the cathode voltage.
You’d probably also want to look at how your amp deals with the heater voltage, to understand if the adapter is even safe in your amp.
Stick to EL34’s/KT88’s there are a lot of options, I actually prefer EL34’s but a lot depends on the circuit.