In my country, the ISP rents you a modem and router. I told them I had my own modem and router during setup and my monthly cost is slightly less than their advertised price.
I am fortunate that my ISP gives me a routable address, but it is still only dynamic and may change a couple times a year. I would have to pay for a commercial plan if I want a static IP. Some other local ISPs use carrier grade NAT, but you can still request a publicly routable static IP with a business plan. Maybe you can ask your ISP for that?
Yes. This is home-made out-of-band management, like HP’s iLO, Dell’s iDRAC, or generic IPMI. Not only is it a virtual KVM (keyboard/video/mouse), you can pass the host’s power button through this device so you can remotely power on or reset a hung or powered-off system, or mount and boot from a virtual floppy or ISO to completely reinstall the remote system.