Welcome to the forum

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My questions are:
Can I use the NHS with my biometric permit?
Can I work/study?
How do I get a NI number?
You will not actually have a biometrics permit, because they are only issued to people who have applied for a visa from inside the UK.
Your visa will be a sticker in your passport, not a biometric permit... but when you apply for your next visa inside the UK in a couple of years, it will be issued as a biometrics permit instead of a aticker.
But, in answer to your questions:
1) Yes, you will have 'free' access to the NHS from the day you arrive in the UK. You just need to register with an NHS doctor
2) Yes, you will be allowed to work or study from the day you arrive in the UK (however, for studying, you won't qualify for the lower UK fees until you have been in the UK at least 3 years AND you have permanent residence (ILR).. but it takes 5 years to get ILR, so you will need to wait 5 years to qualify for home fees).
3) You can find out how to apply for an NI number here:
http://www.hmrc.gov.uk/ni/intro/number.htm