IANAL and not a startup founder who incorporated in the US, but answering as someone with some general knowledge on the subject and some experience incorporating on different jurisdictions (with the usual disclaimer that this no legal or tax advice).
Generally, you can 'incorporate' in the US as a non-US person. I put that in quotes because LLCs are not technically corporations (US LLCs are 'unincorporated associations') so normally the term used is 'organized' or some such. You can also incorporate proper, having a C corporation in the US. As a non US person, what you cannot do is be a shareholder of an S corporation, which is a special type of corporation that is pass-through for tax purposes (in short, meaning that the shareholders pay tax directly instead of the corporation itself being taxes).
If you want to go down this path, it's good you're looking for a good lawyer, as they can advise you on the best way going forward. That said, there are many online formation services that take care of the paperwork for you for some nominal fee (say between $200-$500), and some jurisdictions that are popular for non-residents are Delaware Nevada and Florida. As far as I know, for Delaware this is partly because of some perceived tax benefit, plus it being a place with a good reputation for doing business.
For question number 2, if you'll be working from outside the US, I don't see how it can be an immediate problem, since immigration laws that restrict you from working usually would apply when you're in US soil. As far as I know, there's no restriction on working from abroad, so long as you otherwise follow the law. You could probably even be an employee of your LLC in your local jurisdiction.
You don't say why you want to incorporate in the US and maybe you have good reasons for this. However, some of the negatives are that you'll likely complicate things for yourself, as now you'll have to report to the US authorities (taxes, etc.) in addition to to your local jurisdiction, you'll likely need US and XX accountants, etc. Plus, LLCs are sometimes problematic from outside the US, because they're not really corporations, which from what I hear can cause issues with things like opening local bank accounts, registering with local authorities, etc.
Under transfer pricing rules, you're likely not going to get any tax benefits from operating a US entity, and you could if fact end up double- (or triple-)taxed, depending on where you are and where your clients are and the tax treaties between all those places.
Since you say you're not in the US and mention EU-clients, maybe consider incorporating in some EU state instead?