Yes. An 802.11n card should be backwards compatible with 802.11g/b. If you have a router that supports 802.11g, then the 802.11 card will work. But you are not gaining anything by using the n card with a g AP. The card will simply function as a 802.11g card.
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.
When you say "connect it to the wall jack", you have DSL connection or cable connection? Does cable connection allow the use use of a wireless router and card