Shouldn't the other parameters have exactly the same requirements?
Certainly not all other parameters. The vendor could set requirements for $e$, but they would be different. For the other key parameters, extra requirements would be unusual, and there could be a valid technical reason for them only if you are using gear (software or hardware) that the ...
Can the openSSL random number generator be considered cryptographically secure?
Does it impact the quality of randomness to use a broken cryptographic hash such as MD5 for random number generation?
Not as far as we know. The question you link does, in fact, cover this topic adequately. To quote it to answer your question:
And actually, ...