Just to pile on to this: It's a little known fact that you can have multiple work permits issued. But it's a well-known assumption that you'll most likely be working illegally at some point in the application process. And it should go without thinking that you want to keep your work permits as far apart from each other as humanly possible; in other words, never let the right hand know what the left hand is doing.
Ferris, I'm going to keep this as simple as possible.
Get a side job. The rest will take care of itself, i.e. your secondary employer is on the hook for the work permit. Do your best NOT to involve current employer. It may be inevitable at some point, but just keep it to yourself for now. Besides, if you're not making any dough, your current laoban has to be aware of that. One would hope...
You heard correctly, although they might not tell you to your face. It is a pain in the ass for some of these people. Anything that involves a bit of effort and (perhaps more importantly) a paper trail back to the laoban is frowned upon. I don't blame you for wanting to be above-board about everything, but the best Taiwan advice I ever received (and regretfully dismissed) was to "Stay off the radar". I'll spare you my APRC trials and tribulations.
You wouldn't be obliged to register...unless it weighed so heavily on your conscience that you couldn't sleep. I'm not going to suggest you do anything illegal or unethical, but the fact is, the whole work situation in Taiwan is a massive gray area. My best suggestion is basically my M.O. in Taiwan: Do what you want until somebody tells you to stop.
Many, many people on this board and in Taiwan in general are living if not entirely in a gray area of legality, then dipping their toes in the water every so often. For some, it's unavoidable. Did you work while your ARC was being processed? Yes? OK, you've been in the gray area, too.
Online gigs are awesome because it's completely between me, the client, and PayPal. Don't ask, don't tell. No harm, no foul.
Again, kudos for being straight-up, or at least wanting to be on the level. Sometimes it doesn't help your cause here. Anyway, if I were you, I'd find the gig first, and then sort it out. There is absolutely no harm in looking for more money.