"It is three o'clock."
Is the particle は optional after 今？ Does it depend on how formally I'm choosing to speak?
Yes, it is acceptable to drop the particle は. The comma in 「今、何時ですか」 for example, signifies a pause for where the particle would have been.
Dropping particles in general is less formal, but I believe the difference is almost negligible when it comes to time clauses.
This is stupid, but what if someone named Sanji were to introduce himself, and people would think he was telling the time? Ex. "さんじです", and then someone would say something like "いいえ, 今は一時です".
In writing, you would be able to tell the difference, but not when spoken (at least not without any given context), which I thought was kind of funny.
Essentially yes. But be aware that in Japanese (and in this case, in English too), the subject has been implied.
If the subject is specified in Japanese, it looks like 「時間は三時です」 and the structure is "the time, three o'clock is"