Do people use the phrase, " give them my regards," Or "My regards to [whomever]"? Do they speak that way in Great Britain?

Yep, I'd use it especially when talking to older people or people I'm not too close with. For everyone else it would be "send them my love".

I don't use the phrase, but I know people who do and it's okay with me.

Not often in speech its commonly thanks - but I alwasy sign off enails best regards so ...

Not so much in everyday speech, but in formal writing, certainly.

We also say 'regarding' or 'with regard to' when referring to a particular subject.

I would use that phrase.