Poor Germans! It’s true, max. 3 months per employer.
http://www.boca.gov.tw/ct.asp?xItem=3745&ctNode=783&mp=2
However, they should not work for the same employer for more than 3 months. In addition, holders of working holiday visas should not work for the entire one year of their visit.
I don’t know about the childcare market. While you’re here, I suggest reading this thread for general information about the WHV.
You may be qualified to work as an “educare provider” i.e. kindergarten teacher. I can’t say that for certain, but it’s possible. See here for a long, complicated explanation:
Short version:
You don’t need a work permit, so the question is just whether or not your license for nursery school teaching is acceptable in Taiwan. (Teaching a foreign language class would still be illegal, but it would be the kindergarten’s problem, not yours.) The funny thing is, if you ask the education authorities they might try to avoid answering the question due to suspicion that you want to work as a language teacher (which, again, is illegal for kindergartens but not for teachers).
Being paid cash is legal, as long as you don’t evade tax. (Keeping paperwork makes it easier to prove what you did or didn’t get paid, of course.) Depending on what time of year you arrive in Taiwan, you may be a “resident” for tax purposes, which will almost definitely result in you paying less tax.