Note that I am not in STEM/CS, but I have many friends who are. My answers are based on what they have told me, so I suggest getting a second opinion. Also note that most of this is answered for someone intending to be a professor. Post docs and grad students might have different answers.
1. There is a very high likelihood that you will "drop of the face of the earth" in terms of research if you go there for a long-term professor gig. You will most likely find that the quality of your everyday peers will be less-than-stimulating academically compared to your everyday peers in a top US program. Note that there will be exceptions, but they will be exceptions. Japan is fine if you think that you will be a third-rate or lower researcher in the US (i.e., no publications that will be highly cited either for broad topics (first tier) or niche topics (second tier) -- those are my terms that I find convenient for differentiating academics). If you just want a taste, one option is to go as a post-doc. Aim high in terms of school prestige if you do this, and try to get on a project in which you can do your own thing or that has a topic that you love. There are some schools (or affiliated research orgs) that do international-level STEM research and definitely want western researchers on their projects.
2. Funding varies, but unless you are Japanese or highly integrate yourself into Japanese society, you are very unlikely to get substantial funding directly from the government or industry. As such, the best way to get funding will be from a sponsoring professor in Japan (this is not uncommon) or piggyback on something from your US contacts. Some schools (e.g., Aizu) seem to have a good gig going in terms of funding for their foreign faculty, but I don't know who is pulling the strings (e.g., in terms of actually applying, etc.).
3. It varies by research group. In general, it works as a hierarchy based on experience on the team. Special treatment for exceptional students/researchers is rare unless they are a hired gun (usu. a temporary gig). Post docs might be a slightly different story. It really depends on the field and the school/professor, but I encourage you to set your expectations for respectful treatment to be very low.
4. Not so possible in Japan... sort of. Basically it is possible to bounce around the Type A schools in the "Top 30" (https://en.wikipedia.org/wiki/Top_Global_University_Project) as a Japanese professor. There is a pecking order within those schools (usu. Todai, but sometimes others in specific fields), and grad students typically filter down (i.e., the best usually stay at their school on a space-available basis, and the others take a step down the prestige ladder within the Type A schools). It's tougher to move from a Type B school to a Type A school (but possible with stellar work and/or the right connections). I think it's almost impossible to move from a non Top 30 school to a Type A school regardless of who your advisor is. Note that moves as a professor will likely require strong personal contacts at the target school as well as strong bona fides. If you are coming from the US, a recommendation from a very well known international researcher will likely get you some sort of a job if you work the right channels. I can't tell if you're asking this as a potential grad student in Japan (don't do it if you want to come back to the US) or as a US grad student wanting to go to Japan, or something else.
5. Professors often have funding for conferences. My school had a "research budget" for each professor for such things. There was also a departmental general fund that could be tapped into for special cases (not routine conference attendance). Other schools handle it differently. Regardless, there is money there.
5a. Project failures. Hmmm... maybe "failure" wasn't the best way to phrase it. First, any research that has a legitimate chance of being published in a western journal can 100% be published in something in Japan. As such, it's not really a failure if you have a paper. Actual failures can come in two forms (that I have seen) in Japan. One form is just not getting results in a series of experiments in a given time frame and not having anything "meaningful" to publish. A second sort of failure is simply not publishing regularly -- for example, it is often considered poor form to work on something potentially substantive for several years and not publish a few small (and probably meaningless) things in the interim. The unwritten rules on this are pretty clear. Frankly, I think this is why some people "drop of the face of the earth" when they go to Japan -- they get wrapped up in these limited-scope papers and forget how to move towards more substantive research. The myopia is real.
Misc. I read a couple of your other comments about "target" schools and whatnot. Be careful thinking that things like "top 3 publication-wise" mean anything in Japan. Your school may be a big deal in Japan or at the specific school you apply to, but the reasons will often be due to less academically substantive reasons and possibly due to something random (e.g., someone in the department studied there). Also be careful about being too proud of what you've done in terms of research/publications -- the frames of reference are different.
Let me know if you have any other questions.
