Do I have to take the host for an outing if I've signed up as "Stay Share Play"?

You have no obligation whats o ever to share a sports session with your host. It's entirely up to you, on whether you're willing and available and on whether your host is too.

However, if you've signed up as Stay-Share-Play (see explanation of Stay-Share-Play level of interaction) and so has your host, then he/she will most probably want to share a sports session with you. They might be disappointed if you can't join him.