I don't think your wife needs to have a tax paying job to qualify for her 10 year visa. However, I would have thought any application for ILR would be looked on more favourably if she can show she has integrated into british society/way of life. Taking a job and paying tax would show that commitment. She can apply for any advertised job , it doesn't need to be one advertised via the jobcentre

I am a bit puzzled as to why you are paying your wife your carers allowance. Surely as man and wife you are the "same firm" and therefore you're equal?

Your wife will need to apply for a National Insurance number before she is able to work, if she hasn't already done so.