Re: Internet via LAN
no, you don't have to buy a router, but it will help to bring the second computer up to the speed of the first computer...
(I assume you mean that the second computer seems slower than the first, or have you actually measured this?)
a router will balance the connection between the two PC's, that is to say it won't give you any more of less bandwidth, it'll just share it out better!
(i.e the PC that currently has the faster connection will seem to slow down, whilst the slower PC will seem to speed up).