I formed a single-member LLC in 2021. Do I need to file a separate tax return for the LLC even though it's a "single-member, LLC"/"disregarded entity"?

 No.  Only if you elected  it to be an S Corp.  Then you need to file a separate business return.  Otherwise you file it on schedule C in your personal 1040 tax return.

A single-member LLC that is a "disregarded entity" does not file a separate Federal tax return; you file its taxes on Schedule C same as a sole proprietorship. (Some states may require a separate return.)

No. Any income and expense is reported with your individual tax return on Sch C. 

You do not need to file a separate income tax return for the LLC unless you made an election for the LLC to be treated as a corporation for federal income tax purposes (which appears to be highly unlikely based on your question).

Otherwise, your LLC is disregarded for federal income tax purposes and, in essence, is treated the same as a sole proprietorship. 

Therefore, you would file a standard Form 1040 with a Schedule C (assuming your LLC is engaged in a trade or business).

Just for clarification, the phrase "disregarded entity" comes straight out of the relevant Federal regulations; it refers to single-member LLCs that did NOT elect to be treated as a C or S corporation. (Multi-member LLCs that do not make such an election are treated as partnerships.) Except for some S corporations (they have more flexibility in paying the owner's Social Security & Medicare taxes), it is generally NOT advantageous for an LLC to elect corporate tax status due to "double taxation" of C corporations.