Let's say I have a network where I run pfSense as a router. I want to be able to VPN in to my network when I'm away, but also want to make all outbound traffic from my network go through another VPN, so as to not have to configure each device individually. In this scenario, pfSense would be both client and server. Is this possible?
Yes, that's most likely possible.
Most VPN protocols like IPsec or OpenVPN are originally peer to peer, with client-server functionality often added as an option. Client implementations can usually only handle the client side, server implementations should be able to do peer to peer as well.
That said, all you need to do is configure the desired VPN links and set up routing between them.
Yes, most firewalls can do that.