Since I used to live in Japan as an international teacher, I'm probably the closest you'll get to an expert on this forum.
Some things to note: Japan is known as a premier destination, which means there are way more people applying for positions than positions available. Also - it becomes more challenging to get rid of an employee in the Japanese international school system once they have been in Japan for longer than 5 years, which means that people NEVER leave.
As such, there is probably 6 - 10 positions in the whole of Japan that would be at a school you might want to work at. That's not jobs each year - thats in total! With most of those teachers being in the country 10+ years. How do I know? I know them!!!
Now I'll try and answer your questions below
(1) Wages. Consider 400 K a month after tax as a good wage for an international teacher. Note that wages have been stagnant and benefits depleting over the past 20 years.
(2) Benefits often include original flight to the country (out of country only), some type of transport allowance and the big one is a housing allowance. This changes depending on the city and school. Local hires don't get a housing allowance but do normally get a transport allowance to work. To put it simply, benefits at almost all international schools in Japan have been reduced due to more teachers applying for positions than they have positions. Most schools grandfather current staff.
A good example is that old staff used to get an amount of volume for moving allowance (say something like 2 cubic meters in the first 2 years, increasing by 0.5 cubic meters a year until you reach 3,5 cubic meters. Now - they give you a flat cash amount.
(3) You've mentioned the main schools in Tokyo, apart from TIS which I believe does not have a high school. Outside of Tokyo, there is really only Canadian Academy in Kobe along with Osaka Senri. There are a couple of other schools such as Fukuoka (low pay) and UWC Karuizawa (yay - boarding school in the middle of no-where) which might float your boat. There are also a few good christian schools you have not mentioned, but the pay is rather poor.
(4) In general - stay far away from these chain schools. Most of them are for profit with education playing a far second level of importance compared to profit. There are a couple of good ones (Chadwick, Dulwich) but I don't know if they are in Japan yet.
On getting a job in Japan - it's more really about who you know than what you know. There are a couple of agencies which people tend to use. I use Search Associates, which costs money but allows you to contact schools directly. They also have confidential references which play an important role in getting teachers jobs. Some other options also include Schrole and TIE Online. You can also join free agencies such as Teacher Horizons and the like, but expect people to be contacting you about jobs in China or the Middle East continually. These agencies get paid to find you a job - any job!
On your employability: this is a hard one to say, but your current experience will not put you at the front of the list (even with Chem - I've taught SL/HL DP myself along with ESS). It really is this competitive at the top tier schools
There is also lower tier schools which range from 'ok' to 'get me out of here'. It's common for teachers who have proven themselves at the lower tiers to make their way to the higher tiers. Also note that when it comes to jobs, schools will often post on Search Associates before their website. For lower tier schools, it's the other way around as they don't want to pay the employment fee of a few thousand dollars.
A community which is probably better suited for these questions is here: https://www.internationalschoolsreview.com/
Do note that they can be quite opinionated on the public forum but some of the people know nothing but act as though they do.