You only can use it without one if your upgrading to windows 10. Most systems have the key on the motherboard. As far as running multiple of the same windows 10 its not possible unless its in a VM environement.
Yes bitlocker is the whole reason. And i dont think you would be able to access the drive without some advanced tools. It seems like you took out the drive while bitlocker was still active.
Update the BIOS and all the drivers. Try the keyboard on another system see if the problem happens there. Also try another keyboard. These should tell you for sure if its the keyboard problem or something else.
Test individual compnents on another system if you have one that is comparable, or test memory individualy on each slot. Im suspecious of your PSU though but im not sure how you can test that short of getting a new one.