If I set up a Windows 10 PC with a software firewall (TinyWall, which is a front end for Windows Firewall) and blocked all traffic except for a few applications and ports I would be using (Minecraft server and Plex server), would it be safe to put my machine into my router's DMZ? Would the PC, its data, or my home network be vulnerable to outside attacks?
If you router offers a real DMZ then the rest of the network would be safe even if your Windows PC is compromised. A real DMZ is a separate network which has no or only very restricted access to the internal network.
But, what most SoHo routers call DMZ is actually an exposed host, i.e. traffic from outside is forwarded to a single host inside the internal network and not into a separate network. Since this exposed host is inside the internal network and has full access to the network this setup can not be considered safe because the compromise of the exposed system could also affect the security of the other (non-exposed) systems inside the same internal network.
A real DMZ requires that the router itself provides an additional network for such exposed devices which has no access to the internal network. This network should also be physically or logically separated from the internal network, i.e. different usually with a different physical ethernet port but maybe only with VLAN. If the router only offers to forward everything to a single device inside your internal network (i.e. exposed host) you might try to make the best of it with a strictly configured additional router as exposed host which then tries to offer the real DMZ network. But since the router is placed inside the internal network you have to be really careful with the configuration so that DMZ devices cannot access the internal network.
As for the security of the exposed system itself: Even if only the Minecraft and Plex server would be reachable from outside I would not believe in security since in the past there were exploits for Minecraft server and exploits for Plex.
Apart from that most attacks against normal systems like yours are not caused by direct hacks against the system from outside but by the user inadvertently putting malware on the system by browsing (drive-by-downloads) or reading mail and opening links or attachments (i.e. from phishing mails). And the simple Windows firewall will in no way protect against this. Antivirus and similar will help better against this malware but also cannot achieve full coverage.