Gigabit Ethernet switch under Megabit router

Gigabit Ethernet switch under Megabit router

Suppose I am very poor, which I am, I can only afford a 10/100 router and
a gigabit switch.
If I am to connect everything under my router which runs at 10/100M,
everything will of course runs at 100M.
Now my problem is, if I am to connect my stuff like this:
router w/ Internet connection
- Gigabit Switch
- Computer 0
- Computer 1
And both of those computers have NIC capable of doing gigabit Ethernet.
My problem is,
a) Will they get DHCP IP from my router?
b) Will they communicate at gigabit speed?
(I am quite poor as it comes to knowledge about Internet switch and what
they do, if anyone is to elaborate that further. It's much appreciated.)