If you are being offered a salary of 10 million yen per year, you will have plenty to live on. Average wage in Japan is half that, and entry level wages for teachers alone are 1/2 that.
Tokyo expenses are similar to those of London or New York City.
www.pricechecktokyo.com
If you don't know how much 10 million yen is, learn to use a currency converter, or to check your local newspaper daily for the exchange rate. At today's rate, that comes to US$87,000. You will have to pay taxes on that back home, by the way, as it is over the ~$80,000 upper limit for tax exemption on money earned abroad. So, you may get double taxed (here and in USA).
Weekend travel is what you make of it. An average hotel room costs 5000-8000 yen per person (roughly $50-80). Train tickets may run any price, depending on where you go. Use
HyperDia | 乗換案内 路線検索 時刻表 旅費精算 通勤費 ダイヤ作成なら (click on the English link if you get the Japanese site) to do some calculations on time and costs of traveling here.
Roppongi is a haven for foreigners. Boat people, as my friends and I used to call them. It is an entertainment sector of Tokyo, and prices for living are expectedly high there.
For issues related to women and women's clothing, I'd say go to
www.being-a-broad.com or read the companion book. You're going to find it hard to get clothes here, that's for sure.
Dog. Leave him at home. It will be hard on him during the flight, and as much as quarantine measures have relaxed, finding a place for you to live will be hard, as most apartments don't rent to dog-owners, whether they are foreign or Japanese.
What sort of job do you have lined up? Are they arranging housing & visa?