Can we add a second teacher to the class? Or transfer the class to a new teacher?
No, sadly not. Those two features have been requested many times - hopefully Duolingo will implement them soon. I think Duolingo must be aware aware that the inability to have a classroom belong to two teachers means there are some schools that won't be able to use the system. I think they must also be aware of the admin burden that the inability to transfer a class puts onto schools. It also isn't possible to upload a csv file to create / move students - that's also been requested before, and will hopefully be added soon.
It's early(ish) days for the Duolingo for Schools system, and there's a lot of stuff that a teacher needs that hasn't been implemented yet. Let's keep hoping...