2. Yes, although I forget all the boxes you have to tick in OS 9 to make it work. OS X will find it and get you on the net by magic.
1. First thing you'll need for both of those machines is RAM. I'd suggest 192MB (128+64 or 3x 64) as the minimum you'll want to try OS X in, and stick to 10.2.x. Once you get 10.2 running it'll automagically update to 10.2.8 once you're on the net.
I didn't find it particularly piggish for the six years I ran my beige G3/300 with 10.2, but these things are relative. Bumping up the RAM as much as you can afford makes a huge difference - above 256 and you'll really start to notice it. At 512 and above, the speed on mine was fine.
Unfortunately those machines are both picky about RAM. The iMac takes laptop SODIMM SD-RAM, and the beige takes desktop SD-RAM DIMMs. In both cases, 256MB sticks must be double sided (16 chips to a stick, called "single banked"), which are a bit pricier than single sided 256s. 128s and 64s you can pick up for nothing, or scrounge from old PCs.
Speed wise they'll both take PC-66, PC-100 or PC-133. No other (DDR, PC2300 etc) will work.
On a budget, I'd suggest sticking 3x 128s in the Beige. Or I'd suggest go straight to memoryx.com, ramseeker.com or eshop.macsales.com - select the machines from their menus and you can be sure you've got the right RAM - and buy at least one 256MB stick for each machine. You can make up the slack with 128s and 64s until later.
Clip off the useless fan guard under the power supply /edit/ in the beige / and you won't have to fuss about low-profile DIMMs for the last slot. Note I said "clip" not "cut" - don't use a hacksaw unless you want to spray your motherboard with random conductive fragments. I used a hefty pair of side cutters.
There are three slots in the Beige. Two slots in the iMac; one on top of, and one hidden under, the CPU card. The bottom one has to have a low profile SODIMM.
Do the RAM upgrades first before you start to think about CPU upgrades.
The beige will take ZIF G3s and G4s up to 1GHz - expensive - but there are bargains for used pulls around 300-500MHz.
The iMac will only take a 266 or /edit/ 333Mhz / iMac CPU board from another tray loader. However, there's other nifty stuff you can do to them, like adding ADB, a floppy, and Mac serial port.
Both machines will often overclock nicely; on the Beige it's just jumper swapping, on the iMac it involves soldering tiny surface-mount resistors.
Have fun and let us all know how you go.