Yes. It is fine to say this.

Yes, it's correct. The combination of "come" or "go" followed by the base form of the verb is used mainly in American English. It's less common in British English - we'd be more likely to say "Come and have coffee with us". The use of the AE construction is becoming more popular in the UK, though. There's a reality TV programme called 'Come dine with me' which has probably done a lot to popularise the dropping of the 'and' in BE.

Yes this is correct, but you could also make the sentence sound better by adding, 'and + a comma after come''. . . "Come, and have a coffee with us."

Yes, it is a grammatically correct invitation, but you need a comma after "come": "Come, have a coffee with us"