Every tier runs the same search against the same registry data. What changes is how many domains you can check at once.
One domain, once. A pattern making 1,008 combinations on a free account checks the first 1,000 — the rest are unchecked, not unavailable.
Yes. Every result is an RDAP lookup made at the moment you search, streamed into the table as it resolves.
No. Available names link straight out to GoDaddy, Namecheap or Dynadot. You buy wherever you already have an account.