1. Money can be gathered in many ways. In game time, the fastest is stealing (or more violently seizing) stuff from different locations and selling them. Theft in itself doesn't cause heat accumulation. Out of the money raising methods, the legal ones aren't exceptionally strong, but by making disguises and selling them in stores you can make a fat profit with a good tailor. If your character can outrun cops, try selling brownies. Prostitution is good also, but carries the risk of getting caught in a sting operation and lowering juice for characters with low street sense.
Credit card fraud does generate heat, if done long enough. If your fraudster is a good hacker, you might want to try Hacking. It may generate money even faster than CCFraud because you get rare files sometimes upon success.
And finally, lowering heat can be done by doing community service. It's not very fast but still works...
2. Guard dogs have a bonus for spotting you through disguises, to the extent that even skilled Liberals with the correct disguise are often spotted. I don't think the dog bonus applies against your stealth, so the sneaking approach may still be valid (I must check that from the source). But yeah, luck counts.
3. There's no longer any reliable and fast way to get legal juice. Most legal (or semilegal) actions have a juice cap, so they won't increase your juice past a certain amount. The most significant cap is 200, as I think that's as high as you can get through the 'A'ctivate screen. To get past that you need to kill, terrorize and release prisoners.
There are a few special occasions that may raise juice without doing anything criminal. If I recall correctly you get some (10?) juice by having an epic court defence.