On 1, I would not just presume that the tax will apply only to your second residence - the amount needed by the government is billions, not tens of millions, so they will need to look at raising thousands on average from each of the 2bn households in the country. If I were planning and looking at the finances, I would expect to have to pay at least 0.5% of the value of all your properties each year and perhaps as much as 1%.
On 2, I'm not an expert so I'll leave it to others to answer for sure, but I think the banks sort this out at source, so you probably just need to tell the bank.