From my belief the answers to your three questions are 1. Yes. 2. Yes. 3. Not the government pension, but possibly half of any company pension you may have. Check with your private pension company (if any)
So her income would be her company pension, which should increase yearly + her UK state pension which will not increase after she starts to take it + any widow's pension she may be eligible for from your company pension, which should increase.
In my opinion buying extra years for the state pension is a good investment, but check out the details.