It's a factor, but not the only one. I wouldn't mind: it's already a win that you can create extra authority from the old domain.
In short term, it's more interesting to keep the old domain name. In the long term it's more interesting to have the new more relevant one, kickstarted with (301) links from the old one.

It's a preference, there's no right or wrong here. If going for the long term, relevance is my number one choice. You should see the old domain as a gift to kickstart it.

Depends. If your domain name isn't that relevant for you business, I would buy the new domain name and do a 301 redirect. That way you can still profit from the authority of the old domain name. You won't get everything, but it will still kickstart your new domain.

Relevance is everything. You don't want to change domain names later on.