1. If you are legally resident in Quatar, you can choose to have your application processed at the visa office responsible for Quatar and if your citizenship is from somewhere else, you also have the choice of getting it processed in your homeland. The processing time depends on where you get it processed, see
http://www.cic.gc.ca/english/information/times/international/05-fc-spouses.asp Just wait until you are sure you have lost your PR until you apply for a new one.
2. You do not have to pay taxes in Canada when you are not living in Canada. Your wife can file taxes stating that she's not making any money, probably better if she did to establish her residency because with only 5 months a year, she is not legally resident in Canada either and doesn't technically qualify for health care. Maybe because she's never gone for a full 6 months at a time, she would be ok. It's another matter if the health care system will be watching their movements. In most provinces, you get a health card after a 3 month waiting period. In some you can get it right away if you move from overseas. If you hang on to that health card, the health care may not know you have left the province or for how long.
3. If you want to keep your PR without staying in Canada for 2 out of 5 years you have 3 ways to do that. You are working for a Canadian company that transferred you to another country OR you are accompanying your wife who is a PR and is working for a Canadian company that transferred her to another country OR you are accompanying your wife who is a Canadian citizen. I don't think any of this applies to your situation. If your wife spends 5 months a year in Canada, she will not lose her PR.