Yes, singular "they" is grammatically correct — it has been used in English for centuries to refer to a person whose gender is unknown or unspecified, and it's now also standard as a gender-neutral pronoun for a known individual. Major dictionaries and style guides (including Merriam-Webster, the Associated Press and the Chicago Manual of Style) formally recognise it.
You've almost certainly used singular "they" without noticing, whenever the person's identity or gender isn't known:
This usage dates back hundreds of years in English (writers including Chaucer and Shakespeare used it) — it isn't a modern invention, despite sometimes being taught as an error.
More recently, "they/them" has also become standard as a personal gender-neutral pronoun for a specific individual who uses it — the same word, just referring to one known person rather than an unspecified one:
Grammatically, this works the same way "you" already does in English — "you" is used for both one person and many people, and the verb doesn't change ("you are", whether one person or a group). "They" now works the same way for a specific individual.
No — singular "they" takes a plural verb form, just like "you" does: "they are", not "they is". This might look odd on the page at first, but it follows the same pattern English already uses for "you", which is grammatically singular in meaning but always takes a plural verb form.
The use of "they" for an unknown or unspecified person dates back centuries in English, appearing in the work of writers like Chaucer and Shakespeare — it isn't a recent invention.
Yes. Merriam-Webster, the Associated Press Stylebook, and the Chicago Manual of Style all formally recognise singular "they", both for an unspecified person and as a personal pronoun.