If you get fired and apply to another job, can they find out that you got fired and for what? Has anyone gone through this?

Most employers will only confirm dates of employment and not terms of departure.

M1 is right. There is too much liability for a firm to disclose this. The new firm may ask why you left the last place and that is where you can take A1's advice if it seems appropriate. Personally I would not disclose too much, just say it wasn't a good fit and move on to how this company will be a better one.

Oh - and if it's criminal or a matter of public record - yes! Remember there's likely a probationary period. I don't know about others - but I can smell BS in a new joiner. Don't raise suspicions and have people start digging. On the upside - being up front as to how you tried it on with your managers wife at the holiday party and got fired will mean no one will suspect you of the stuff you hide...... ;)

You also need to be aware that depending on your specialty it can be a small world. I know my counterparts at the other firms and I will talk honestly if someone calls me personally and asks if I know XXX person. Tread lightly about fibbing about your departure.

Yes. If you did something wrong, own up to it. Confess contrition. Explain how you've matured. They will find out otherwise.

A2 - I guess it all depends what OP has done!

HR can only ask HR whether you are fit for employment or not

Think of every tactic you've used for undergrad recruiting. They can use all of the same. It's not uncommon for Wall Street for example to call up their buddies in Consulting and ask how XYZ was.