Some customers are more confident buying your product if you are a corporation, but on the other hand while you are bootstrapping you still need some cash coming in, and one way to do this is to work for an employer.
Would I be able to register my company as LLC and run it while still working for an employer?
Yes. There is absolutely no law against that, at least in the US. I'm actually doing that myself. Just keep in mind that if you do this you won't have as much time to dedicate to your own business. It's a tradeoff.
A side benefit of doing this is that your employer will withhold your taxes, so you shouldn't have to worry about messing with estimated taxes.
It depends on your employer. For example, with Oak Ridge National Laboratory, if you are employed by them, you need to get permission from your boss before you can work elsewhere. I think if you are in a classified job, or a job where there can be ethical conflicts, such as in a role where you make purchasing decisions, then you may find that the business that can supply to your company may be a problem.
But, if you don't have a policy in the corporate handbook restricting you then you should be free to do on your time what you want.
This also may be a problem if your company has first dibs on any IP that you do, even if on your own time, with your own equipment.
So, look carefully at the employee handbook then see if any of this applies, otherwise, start one up and good luck.
Yes, however check out your state requirements to make sure you're allowed to do single member LLC's. And, as JB mentioned above, check with your employment obligations to avoid conflicts.
Look for a non-compete/non-competition clause in your employment contract for applicable restrictions.
You can have multiple LLCs, however make sure your business is not in violation with your employment contract.