I'm going to assume you're in Taipei and have my response geared towards Taipei locations and stores, but next time don't forget to add the city you're in. Taiwan is more than Taipei!
Secondly, there's some things that you pay a premium here for convenience. A lot of name brand things are going to have a nice hefty price tag on there and sometimes, you can't do a price comparison to Canada or the US. Just think of it as, you're paying extra for the convenience of having it....right now.
I personally stick to the name brands such as Gap, H&M and Uniqlo. They have a multinational corporation quality assurance and control behind them. I've yet to have any major issues with the clothes I've bought from them.
1 - Search FB for second hand children's toys groups. Some parents just want to get rid of things and you can get good prices for them. Kids will outgrow them so, no point in buying something too nice! Let me know if you prefer brand new, I know a few stores.
2/3 - Sport? As in running? Cross training? Golf? Badminton? You can head over to the Neihu area near Costco, there's a building with Nike, Adidas and Puma outlet stores. Usually a good 30-40% off store prices. Dress shoes really depends on what your budget is. If yours or the husband's feet are foreigner sized, you're better off buying them from home since it's tough finding sizes above 11US or 43-44EU.
4 - I have never owned a front loading washer, but I have been using a Japanese name brand one for awhile and as long as you clean it with the right cleaning detergents, it shouldn't really die on you. Hermana @Icon can elaborate on this, she knows the ins and outs.
If it's a no name brand, I suggest you buy a new one. You can take a gamble and buy second hand from the FB group, "Taipei Buy,Sell,Trade" some of those being sold are from expats that have been here for a very short time and thus, the units are barely used.
5 - Shoes for 2 and 4 year old, I would definitely buy from the night market or from Canada. No point in buying anything name brand here since they'll just grow out of them in half a year.
6/7 - I can't really comment on this, but I assume there's a lot of different kinds of face cream. I wouldn't go any farther than the first floor of some department stores or some of those Korean make up shops in the Sogo area (if you're in Taipei)
8 - If you're in Taipei, the underground malls of Taipei Main Station, Zhongshan, and also the East Metro Mall (underground mall under Sogo) all have stores that only sell umbrellas. They should range in and around 200-600NT depending on the size you want.