A: Yes, for most people with Hashimoto's it's helpful to take probiotics. However, some people might have adverse reactions to them.
There are some case reports of people with suppressed immune systems, such as people with advanced HIV or AIDS, who may be very susceptible to infections and any kind of bugs. And even the probiotics that normally wouldn't colonize healthy people’s bodies could colonize theirs.
People with Hashimoto's have autoimmunity, which is essentially an overactive immune system, not a suppressed immune system. I think immune system problems could mean a lot of things. It's a very vague term, but in the case of probiotics, I think they're referring to people with suppressed immune systems. That said, some people may react poorly to probiotics, and it could be due to SIBO or potentially die-off.
I have an article on probiotics on my website that might be interesting for you to read.