An odd question for a forum like this but I reckon I can get you to a point where you can either ask better questions or answer yours.
The notion of setting out to work for a non government company seems odd to me unless you are in South Korea or Japan, and both of those may not have that model last for the longest. Can be done though and if you are still in school they may even have a nice intern and scholarship/postgrad program for certain fields. I would usually encourage people to instead find something they like doing and then someone to pay them for doing that than to set out to work for a company. That one of their division/marketing logos is your avatar sets off my fanboy alarm but to each their own.
As far as most vaguely working in silicon are concerned there are two types of company
1) Fabless semiconductor company
2) Companies with a fab
This is changing subtly as the years go on but it holds.
A chip fab is a place which makes chips. These can be all over the world (I once went to Intel's one in New Mexico for instance).
Most other companies just buy in chips or have designs made elsewhere, assemble them and sell them on. In the case of graphics cards they are often even provided with a "reference design". They may customise a few things depending upon their warranty and sales department says but it is nothing too drastic and mostly just changes the colour of the solder mask. Asus has a lot of fingers in a lot of pies but is still mostly the latter by the looks of things. There is a marked difference between sitting in a bunny suit properly doing things and setting up a pick and place machine, not least of all in the education needed to do it (the former may see you surrounded by phd having types, the latter is at best semi skilled labour and the less of it they can get away with the better for them, certainly if you are going to be in a rich country like the US).
Others mention China and Asus are actually from Taiwan which is not China per se (the history is rather complicated) and actually represents something a bit different, though they do have operations in mainland China.
"Asus team job"
Time to look up what "Asus team" is, if anything. I would expect a skunkworks to have a cooler name if it was that but I have been surprised in the past.
I got back nothing. Few people mention Asus team as a sponsored computer game playing team but I only got their internal team building documents, a few news articles where they joined with another company to do something and a few references to customer service.
In that case they likely have a range of jobs from PCB designer to web developer to marketing to sales to warranty to legal to programming (while it may be mostly slapping a logo on something existing it is still something).
Looking at their website https://jobs.jobvite.com/asus
Most of their US stuff they are advertising for now, and showcasing on the site, appears to be the typical frontend you need to run a business outside your original company. If you want webdev and marketing and it has to be asus then it looks like you can do it if you want to move to California.
Other places don't mention US manufacturing (mainly Asia and parts of more eastern Europe, which is fairly typically for most such companies). If you have the capability to move to such a country then more can be said.
Their Canadian branch mentions a repair technician in Ontario. http://promos.asus.com/CA/Employmen...092.964370545.1513607764-553757694.1513607764
Sounds like utterly basic diagnosis work to me (cheaper and better for PR to have someone swap a screen in country than ship it all around the world). That said I don't know if if is the start of a proper repair section or just triage/frontline stuff and they ship it back to Taiwan for anything interesting.
That said I did not find where their design work happens so I will have to leave that one to you.