Exactly like any other LLC. Anonymity is a public-records feature, not a tax status, and the IRS sees the same entity either way. By default a single-member LLC is a disregarded entity reported on the owner's return, and a multi-member LLC is a partnership filing Form 1065. An LLC can instead elect S-corp or C-corp treatment by filing Form 2553 or Form 8832.
No. Federal tax treatment is identical whether or not your name appears on the state record. What does change your tax bill is the state you form in and, more importantly, where you actually do business: Wyoming and Nevada levy no state income tax, while New Mexico and Delaware do tax income connected to the state. Forming out of state does not exempt you from tax in the state where you actually operate.
Yes, if it is a foreign-owned single-member LLC treated as a disregarded entity. It must file Form 5472 attached to a pro-forma Form 1120 to report reportable transactions with its foreign owner, even with no US income and no tax due. The penalty for not filing is $25,000, which is why this is the single most-missed filing among non-resident owners.
A multi-member LLC files Form 1065 by March 15. A single-member LLC reports on the owner's Form 1040, due April 15. A foreign-owned single-member LLC files its Form 5472 and pro-forma 1120 by April 15. Each can be extended, but an extension moves the filing deadline, not the payment deadline.
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Wyoming from $397 all-in. New Mexico from $347 total. 5-10 day turnaround.