Of course, don't counter-offer unless you're prepared to walk away if they fail to meet it. It's a high-stakes game, and sometimes you'll lose.
The amount of time you ask for should fall in line with whatever narrative you've been weaving. If you've given them the impression that you're a sought after candidate with many suitors then asking for a few weeks to wrap up your other interviews would not be out of line. But it's very case-specific, you've got to keep in mind how much pressure they are under to fill that position fast. I typically say I'll get back to them next week.
Run interview processes with companies concurrently. It makes it easier to negotiate because you have all of the other opportunities
Also, never give a number first ;)
If a recruiter asks, I tell them I'm not comfortable sharing it. If they press, I tell them that I only discuss numbers once the company and I both agree there is a mutual fit.
No one has ever told me they will not proceed without my current salary, but if it ever came to that I would thank them for their time and hang up.