First things first - this is a complex area and she shouldn't rely on info from a forum. She should take paid for professional advice (the below is a rough outline of the position - certainly not advice):
1a. IHT is charged on the estate of the deceased. So the question is whether the person they inherited from was subject to UK IHT. The UK charges IHT on worldwide assets for individuals who are UK domiciled or deemed domiciled (ie been resident in the UK for 15 of the past 20 years). However, to complicate matters there is a 1956 double tax treaty that can override this for indian domiciled individuals. However, in this case your client has no exposure to tax assuming the estate has been dealt with correctly (which is the executor's issue not your client's).
1b. UK residents are subject to CGT on worldwide assets unless they are non-domiciled and the remittance basis is claimed. This is free for the first 6 years, but comes with a charge (£30,000-£60,000) from 7-15 years and no longer possible after 15 years as the client becomes deemed domicile. Any gains sheltered under the remittance basis are taxed if they are brought into the UK.
2. Yes unless it's under £2,000 and not brought into the UK. If it's above £2,000 she can claim the remittance basis (subject to the charge) and as long as it is not brought to the UK it is not taxed. If it's brought into the UK it's taxable as normal.
3. Yes, subject to point 2 - double tax relief should apply with the tax paid in India deducted from the UK tax
4. See 3.
5. If she permanently settles in the UK this will change her domicile and she will no longer be able to claim remittance basis and will be taxable on worldwide income and gains. Buying a home doesn't necessarily mean she has settled permanently, but it is an indication of intent.
6. She will need to declare any funds brought into the UK. If income is below £2,000 and not brought into the UK then she doesn't need to declare anything. If it's £2,001 she needs to declare it all or claim the remittance basis (and pay the charge).