Hey Archimedes is a cool name for a cat, it even reminds me a bit of Acorn computers. My cat was named Black Magic when I got him from the animal shelter, but it suits him. I used to have a stray cat that would visit from time to time, stay the night and have a meal, then leave the next morning. He was lanky, grey and old looking so I named him Gandalf.
pkupcik, those two joysticks you ordered from AmigaKit will only work with 1-button games. You really should look into getting a 2-button joystick/control pad (Sega style) or a proper CD32 pad.
The Competition Pro has four buttons that all do the same thing. It has a switch that disables two of the buttons. I can't think of anything so stupid, why can't that switch change it to 2-button mode? You need to modify these joysticks for them to be any good for a lot of Amiga games. They're more useful for Commodore64 users.
I'm glad you got the accelerator with 32MB RAM. You definitely wouldn't want 16MB if you want to try some of the more recent Amiga games, make sure you download Total Chaos in advance, ready to copy over to the Amiga and extract.
That RGB-VGA adapter will be useless unless you have an old multisync monitor. It will let you display Workbench in 640x480 on a regular VGA monitor, but you won't be able to play any games with it, you'll need to have your Amiga plugged into a TV or something for that.
The PC-Amiga adapter is only for using PC Analogue joysticks with the Amiga, which is okay but there aren't many games that support analogue controls on the Amiga. Look out for racing games that allow you to use the mouse as a control, those games should work (No Second Prize for example).
What kind of display are you planning on using with the A1200? Since you don't have an Indivision, you will need something that supports PAL/NTSC 15khz inputs like a TV or old RGB monitor, or a multisync monitor. Most LCD TVs will work, but Composite can be pretty blurry to use. If you're going to use a TV as your display, I'd recommend either using Scart (if the TV has that input) or getting a RGB-to-S-Video adapter for the Amiga which will give you a much sharper image with no dot crawl or colour bleeding. If you want to use a regular VGA monitor, you can get these litle boxes now that have three inputs (S-Video, Composite and VGA) and one VGA output. You can plug the Amiga into one of these and it will scan-double and flicker-fix the output for displaying on VGA monitors. If you use one of these, again it's better to use the S-Video than Composite, but since you have that RGB-VGA adapter you could plug this in as well as Composite, and just use the controls on the box to switch between Workbench (VGA) and games (Composite). Unfortunately if you were to use an external S-Video adapter, you couldn't also use the VGA adapter since they both plug into the RGB port, but since the PAL 640x512 screenmode runs faster than the VGA 640x480 screenmode and the external box would be fixing the flicker anyway, you'd just be better off sticking to using PAL all the time if you had an S-Video adapter. The only good thing about using a VGA screenmode is you get a smaller mouse pointer.