On my router I can set static DHCP addresses. I do it for all the known items on my network. That way if something unknown pops up I can check it out. I use 3rd party Tomato firmware, so I don't know if I'd call it "easy". DD-WRT is another 3rd part firmware that also has static DHCP. You would have to get a router that is supported, then flash it, then configure it. I don't know if there are any off the shelf routers that have a similar feature. Of course there are web sites that will sell you a pre-flashed router. If you are looking for cheap, you can probably get a supported router for less than $20.
As for port forwarding, any router should be able to do that. But have you considered Teamviewer? It's easy, and free for private use.