Almost always, send the demand letter first. It's cheap, fast, and resolves a large share of disputes. Suing is the escalation when it's ignored.
A firm, dated demand with a deadline and a credible threat of suit gets many debts paid in days. It also strengthens your case if you do sue.
If the deadline passes, file: New York City small claims up to $10,000 (around $5,000 in many other NY courts), or civil/Supreme Court for larger amounts. Filing fees are modest; you can seek interest and costs.
Usually not, but it's smart: it can prompt payment and shows the court you acted reasonably. Some contracts require notice first.
Up to $10,000 in New York City; many other New York courts cap small claims lower (around $5,000).
A letter can resolve things in days; small claims typically takes a couple of months to a hearing.