At a guess this is a hub or switch. most likely hub if no power. but you probably don't care because you don't want to use this device.
The cat5 ports are 10/100 ethernet. Not rated faster.
The cat5e port is 100mbit/1gbit ethernet. Rated to handle gb ethernet.
Suggest the following.
1. If no wires come out of wall and go into the hub/switch then stop. If there are wires check the markings on the wires that go into the "small hub-like thing". If they are CAT5 wires then you may get errors trying to push gb ethernet. If they are cat5e cable you are all set. If only CAT5 decide if its worth the trouble to get this working given modern wireless has speeds > 100mbit. Mostly likely the cat5 will work, but ...
2. Find out where the cables go to in the house. Easiest way to do this is to plug the remote end into a laptop. Then in the switch closet try each cable in another pc until you get a signal. Then mark the wire, move the laptop to a new endpoint and try again.
3. The spot for your router in your house is pretty much fixed by where the cable guy or DSL guy installed it. If it were in the closet that's perfect. If not then see if one of the cables goes from the 'small hub-like thing' to somewhere near the router.
4. Buy a gb ethernet switch. example TP-LINK TL-SG1008D 8port $25 at amazon. It will replace the old box ""small hub-like thing". No point dealing with the old hub. You will need an electric socket to power the switch.
5. Connect the OUTPUT port from your router to a wire going to the closet. Connect that wire in the closet to your new switch in any of the ports.
6. Connect the marked wires into the closet to the new switch
7. After step 6 the cables in the remote rooms that run to the new switch are now working, you can connect PCs to them and get internet access.
8. Any additional ports on your ISP router (often there are 4 ports on a DSL or cable modem router) can be used (this is in addition to the port on the router that you connected to the wire going to the new gb switch in the wiring closet).
9. Wireless from your router also works.
10. All of the devices are on the same subnet, and can stream media, etc. without problem. The switch does not act like another router or wireless bridge and will not mess this up.
good luck. post what you ended up doing