National insurance and tax questions on low pay..?

When you start a part-time job you're often taxed on the umption that this is an extra job - ie you are already working full time! So you pay a higher, not lesser, proportion of tax. You should be able to get your tax code changed, and off the top of my head I don't think you should be paying anything. But you may have to wait till the end of the tax year, next April, to reclaim what you've overpaid.