If your router supports compatibility mode for b-type networks. You need to check the router manual for the specifications. If it supports in the wireless network properties you need to choose to use mixed mode (can have a different name).
The same as with wireless G card and G router.
You need to change the network card in your computer with a wireless one, and buy a wireless router. Need 2 items: wireless router wireless network card
Yes.
If you do not see signal from your router, it means that the signal is not strong enough an you need to get a wireless extender or a better router.
You can use any wireless card for your new computer. The linksys router uses the same standard for wireless internet as all the other routers. Any modern wireless card will do.
Yes.
yes
yes you can
You buy a wireless router, wire the desktop and router to the modem, and use the laptop by getting a wireless card or usb wireless adapter.
No. You can not attach a router and use a standard Verizon Wireless aircard as a wireless access point. This is why they have MiFi which is essentially a wireless access point.
A wireless router "N" has one of the fastest speeds available. It is more costly than the basic wireless router speed of "G". To ensure getting the fastest speed, be sure both the wireless router and wireless card have the same speed. Most laptops with built in wireless cards only have a speed of "G". The network runs at the slowest speed that is being used. An upgraded card may be needed.
A wireless card for laptops is used to enable the laptop to connect to a wireless network. A wireless router is used to keep track of the whereabouts of devices connected to the network. It is part of the network. This is a very basic answer but I think it should be enough to answer your question.