Sorry for the slow reply. I'm pretty sure UC would qualify. It's a state university. Some technical colleges would not. I don't believe physics or math would qualify, but it's hard to say. I've been told that my Comp Sci degree would be OK, even though it is a science degree. If you are doing it with an aim to try for skilled transfer, it would be best to call up the embassy to get advice.
In terms of transfers, being a contractor doesn't count. You need to be a regular employee and you need to have worked for the company for at least 1 year as a regular employee (your contractor time doesn't count for anything).
Honestly, as a contractor, I wouldn't feel bad at all about being clear that you are looking to get work with the Japanese parent company. But it depends on your relationships. Make sure you've been working there for a fair amount of time (either until your contract is almost up -- or at least 18 months) so that they don't think of themselves as just a stepping stone. But it's completely normal for a contractor to be looking for new work. That's why your are a contractor!
Here's how it works in Japan. If you aren't hired out of school, you can only get a job as a contractor. These contracts are year long affairs (and always start on April 1st). Around about the beginning of March the next year, you hear if they are going to keep you for another year. For most jobs, it's pretty clear earlier, but it's never official until sometime in March.
If you've been working with the company for several years and are doing a very good job, you may be made an employee (shokunin). The company has huge responsibilities for shokunin, so this is not a light decision. Also, this is usually one of those "jobs for life", so it's important that you take it seriously too. If you take a shokunin position and leave it (for any reason), you almost certainly will never be offered another shokunin position again -- anywhere. You might even have trouble getting contract work, so like I say, it's a huge deal.
If you want to get a transfer with a Japanese company, then you need to be a shokunin. There is no other way. They can't transfer contractors. If you decide that you like the company and are willing to devote your working career to them, then I think it's OK to discretely let them know. You can say, "I'm looking to work for the main company and maybe someday maybe even be an employee".
I should say, though, that employees of Japanese companies rarely get much say in where they are being transferred. If you manage to get a shokunin position and you Japanese is good enough, there is a good chance they will send you Japan for a few years to learn the business. However, they might just send you right back to the US or China since you have the special skill of being comfortable in those cultures. In fact, it is likely that this will be your big selling point if they go that far. So.... It's maybe not the best option :-)
Trying to find a US company that can transfer you there may be better.
However, by far the easiest way to get to Japan to do engineering work is to get an engineering degree and get a job offer at Rakuten or Line or one of the many other large companies that hire foreigners. Quite a lot of the development in Tokyo and Osaka is done in English, even.
If you just want to live in Japan and don't mind being poor, then you can always get a job teaching English. You need a bachelor's degree of any type (which you have) and you have to be fluent in English (best if you are a native English speaker). Having Chinese speaking ability is probably a massive plus in larger centers because teaching Chinese is getting more and more popular. The main problem is that you only make between $20K and $30K a year.
Finally, for JLPT... Sigh... I have to admit that I've never actually taken it. However, every single person I know (except 1) who passed N1 did it through self study alone. You can find guides in the bookstore. All of them are good for studying for JLPT.
For learning in general, I advocate free reading. Start with manga because it is pure conversational. Best if you can read stuff that is "slice of life" over fantasy. Once you can read manga fairly easily, switch to children's books (like Momotarou, etc) -- the grammar and vocabulary in children's books are more literary and not how people talk, but it's a good gateway into novels. Look up 昔話 on google and you should be able to find lots of free resources. Finally move on to novels for young children (I read 若おかみは小学生！ mainly because a friend recommended it -- very easy to read and there are lots of books). Then work your way up to more difficult novels and non-fiction books and magazines.
There is a listening test in JLPT but no speaking test. If you read some manga or young-kids books and then watch the anime, that will probably be enough to get the listening skills you need.
Use anki (SRS software) to memorize vocabulary. You can also use it to memorise exemplars of grammar -- just grammar examples. Tae Kim's grammar guide (google it) is the only thing you need for normal conversation in Japan. Start with that. Once you have acquired everything in that guide, then start working through the grammar lists in the JLPT guides.
Since you can read Chinese, I would skip studying kanji at all. The only thing you should do is to set up an Anki deck to memorise all of the vocabulary from the kanji -> reading only (and a different deck for memorizing from English -> Japanese reading/kanji). This should give you more than enough familiarity with the numerous Japanese readings for kanji. The other thing you should probably do is to memorise kanji -> reading for lists of common Japanese names and place names. I can easily pass the kanji requirement for N1 and I've never studied Kanji readings in my life -- I've just read a lot of kanji ;-)
2 years is possible. My one friend who took a course did it at some academy in California (I can't remember the name... It's some animal name and "Academy") They offered a 2 year course to pass JLPT 1 (this is before the N-1 was offered, but it's similar). The cost was $60K IIRC :-) But it worked (at least for my friend) -- he went from nothing to JLPT 1 in the 2 years.
The main key is to learn language through context. Try to find example sentences for every piece of vocabulary you need to memorise. Learn those sentences instead. The other main key is that you need to study frequently, but not necessarily for a long time. 3 times a week, no matter how long you study, will result in hitting a ceiling. Best is to study 10-15 minutes every couple of hours, 7 times a week. If it's constant like that, you'll learn very quickly. It's tiring, though.
Anyway, good luck!