Please note that teaching is also very much considered "white collar".
Answers to your questions are as follows, (however these answers only apply to so-called "ordinary foreigners", the situation for those married to ROC nationals is different):
Q: Can I apply for the Visa through the other company before my existing visa gets cancelled?
A: If you are a fast talker you might be able to negotiate it. Technically speaking, you are only allowed to have one Work Permit at a time.
Q: Upon cancellation do I have a grace period because I am changing jobs within Taiwan? Or do I have to come back into Taiwan on a visitors visa, and wait for the paperwork for my new visa/ARC to be processed?
A: As stated above, if you somehow start early on the new Work Permit paperwork, and get all the necessary approvals done within fifteen days of the adjusted expiry date of your now cancelled "original" ARC, you might be able to pull it off. But why not just go to Thailand for a week, go swimming, drink beer, come back on a tourist visa and sort everything out then?
Q: In some schools a clause is in the contract stating that, upon termination of the contract on the employee's part, the employee will need to pay the school X amount of NT$. Is this actually enforced and can the schools actually use this clause in a court of law, or is it a deterent to prevent people from changing schools?
A: Clauses in contracts which are unenforceable in the Taiwan courts tend to be those where you are renouncing your legal rights. This is because those legal rights (such as "freedom") cannot be renounced in a democratic society (sic). Considering the contract terms you have stated above, these appear to represent a fair agreement between two parties. For example, you cannot argue that you were under coercion to sign such a contract, in fact, you signed in under your own free will, and (apparently) you understood the provisions.
Q: Do the Taiwanese authorities deliberatly make the laws awkward and unclear to confuse foreigners?
A: I don't think so. However, I find that the local ROC people do not understand the full scope of all the laws that apply to foreigners. Hence, different people will give you different (and contradictory) advice.
Q: Is there a scam between Thai airways, the visa office in HK, and the Taiwanese government to make money ?
A: I assume that you are referring to the need for foreigners on short term visas to make frequent visa runs to HK to get their visas renewed. This appears to assure the airlines of full bookings, but I think that this is not a deliberate tactic. No doubt some foreigners are especially sensitive to this, particularly if they have a "short term visa".
While we are on this topic, I would note the following curious fact: According to Article 03.01.07 of the Immigration Law, (which is the seventh item of Article 3), "residency" is defined as "living in the Taiwan Area for more than six months." (This is my translation, taken directly from the Chinese. The translation on the Bureau of Entry and Exit's website is incorrect.) In all the time that I have studied the Immigration Law, I always found this definition extremely over-simplified. (For example, in this definition there is no mention of "Purpose of Residency", which has traditionally been limited to five very narrow categories.) Interestingly however, this definition does leave open the avenue for a legal challenge for a foreigner on a short term visa to apply for a resident visa based merely on the fact that he/she meets this "definition of residency"! To my way of thinking, undertaking such a legal challenge would be a very positive development for the foreign community. If you want to do that, we should initiate the procedures in the period of early July to late December of the year in question. Hence, that means before the end of December 2000, (or else we wait until after July 1, 2001).