Depends on the school and the teacher. I know schools where you spent half the time on your knees and only speak when spoken to and I know schools where it's quite casual, good training but a joke once in a while is OK and I know schools that are a social club only and the training sucks. You got to check out the schools before you sign up. Ask for free trial classes too. That will give you a good idea what they are all about.

Karate is the almost identical to Taekwondo(Since it was also made from Karate). You wouldn't learn anything new there but 1 or 2 hand strikes,

TKD is more and leg work Karate is more hand and control hold work. I would go for it, if nothing else it's more tools for your tool box. And how strict lessons are for laughing and joking very not on art but dojo to dojo.

Very diverse. I spent 9 years at a club, and to be honest, it depended on the mood of our sensei. (he was just english, not asian) If he was happy, he would be lenient, but if someone pissed him off at the beginning, we could be put through hell (stuff like 45 mins of non stop fitness work). Excellent teacher though and it's very rewarding. Really does depend on who's taking the class.

I earned my black belt in taekwondo 4 years ago, then moved away, meaning I couldn't continue my training. I've found karate lessons near me, so I'm thinking about taking up karate, though I'm worried that it's too heavily disciplined for my taste. With taekwondo, I was able to have a laugh with friends during lessons (not "messing about" but joking around without disrupting the lesson) Would I be able to do the same with karate, or will the lessons be very strict and disciplined?