Is there a difference between "hitmen" that mobsters use and "Mercenaries"?
Legally yes absolutely, an assassin preforms an illegal task, ie murder for money whereas a Mercenary is, typically, a professional soldier who fights for a national government for financial compensation, making what he does as legal as any soldier in a national army. Morally, it might be a bit murkier.
Is the politically correct term for them "Contractors" or "Private Security Personnel" or something else?
Contractors, Private Military Contractors, Private Security Personnel, pretty much anything other than "Mercenary"
Is it true that a lot of them are ex-military?
Yes, Blackwater (now Xe Services) was founded by a Navy SEAL and aggressively recruits former special forces from America and around the world notably Chile for operations in combat zones. Dyncorp requires people to have served in law enforcement since they train police This saves the PMC (Private Military Company) the expense of training raw recruits and gives them recruits that qualify for security clearances
Is it true that they exist within the united states?
Yes absolutely, Dyncorp and Xe are the biggest, Xe, then Blackwater was actually deployed to New Orleans after hurricane Katrina
Are there any branches of the military using hired mercenaries to preform operations?
All of them use PMCs, private contractors, cook, do laundry, drive supply trucks and provide security for all branches of the US military, this allows the US to deploy the military without having use a draft to fill these jobs
Is it true our government trained many of the Cuban Mercenaries that exist today?
I don't know. I know the US trained a lot of Cubans for THe Bay of Pigs, but I'm not sure what happened afterwards