i think most CCs (visa/mastercard)can be used for foreign currencies. as long as the store accepts visa/mastercard/amex/etc. you shouldn't have a problem buying with your CC. exchange rate when they convert it to peso will be of course dependent on the bank of the CC.
1. don't know, but you should always pay the full amount of your CC bill anyway, right?

2. usual requirements: payslip, ITR, Certificate of employment, valid ID
3. depends on the type of card; higher the card, higher the credit limit and higher the annual fee. most CC companies offer first year annual fee waived but i think usual fee is around 1-1.5k for the silver (entry level) card.
4. depends on the company and how fast you give your requirements. for my case around 2-4 weeks both for citibank and HSBC both of which are my first time CCs. it's a bit faster if you have other CCs since verification is also faster.