I make a ton of stuff in this way.
Just clean it with an abrasive pad of some sort. I usually use a scotchbrite type thing. No chemicals are needed.
As for the islands, get some 0.8mm thick single sided FR4 PCB and some good scissors and cut it into strips. Then glue down with superglue. Do not use hot glue - it'll melt the moment you solder anything!
As for cutting the main FR4 substrate I use the following tools:
1. Scribe for marking it (copper side)
2. Metal rule for measuring (duh)
3. Engineers square for angles
4. Aviation shears for cutting it.
The latter go through the stuff like butter and makes very accurate cuts but requires some practice.
Here's a pile of stuff I made recently as an example of technique:
Try and build bits of circuits in modules, then test them, then assemble them later. Involves less pain then.
I call my technique "manflattan" as it's manhattan style but flat because I bloody hate all those excess bits of lead sticking up plus it's horrid at VHF and above.