Hi Brad,
1. Depends how much you both want or need to use the laptop. The only problem that I can think of is that if you were both teaching at the same time off the school premises and the school had an online record keeping system, then one of you would have to take notes in handwriting about work covered, student presences etc etc and put it all on the computer afterwards. If you were on the premises then they should provide computer access in the classroom or staffroom. The same for online teaching if the school runs online courses.
2. I'm not an expert on this area of the world, but I believe that in China the academic year begins at the start of September, whereas eg Japan it's April. See
here for a range of other countries. That refers to the state school systems, but in my experience, private language schools follow the same system of terms, adding in "summer courses" in the main break. The best time to apply to private language schools anywhere tends to be a couple of weeks before the start of the new term or summer break, when they've seen how enrolment is going and know how many new teachers they'll need. But that only works if you are already "on the spot". Interviews can be done over the web these days, but if you've got to get visas, find accommodation etc etc it's a different story. At that point you'll need to find work in advance, and that's more difficult - which leads to...
3. Can't help you with websites I'm afraid as (as I said) I'm not an expert here. But be very careful. You say that you have no experience. That's fine - we were all like that once. But have you done a preliminary training course? If so fine. Or do you have a teaching certificate from another field? That could be useful. But if not, you're unlikely to find a really reputable organisation which will take you on. Think about it in the same way as you would about consulting a doctor. Every doctor has to get through their first days of work. But would you want to go to a hospital which employed "doctors" with no training, and who had never been supervised by more experienced staff before being "let loose" on patients? EFL teaching is complex, and an institution which doesn't demand qualifications or experience is likely to be of very low quality. Not always, but often, that translates into exploitation of staff as well as students.
However, perhaps someone with more experience of the geographical area than I have would like to reply?