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1 Not unless you let it out.
2 You will need to register as a non-resident landlord which you can read about here. The first £11,000 of your rental profit will be covered by the personal allowance and you will pay tax at the rates set out in 1 here.
3 Use the calculator here. If the first purchase was the £275,000 property, the stamp duty would be £3,750. The second property's stamp duty will be subject to a 3% surcharge and on £650,000 the charge would be £42,000. The stamp duty rates are here.
4 So long as you maintain your non-UK resident status, you will only pay UK tax on your rental income, not your foreign earnings. Take a look at RDR3 on the Statutory Residence Test here.
5 Take a look here for details of letting expenses you can offset against the rental income. See 2 above for tax rate information.
I hope this helps but let me know if you have any further questions.
The stamp duty calculator link is here.