How much would opening and closing a CC hurt my credit score?

I want to accumulate some of the points across the various Chase Credit Cards and buy a ticket for my girlfriend as a gift so that she can come visit me from London.

I currently have
1. eBay Credit Card. Limit $6500. Owe $50.
2. Chase Freedom. Limit $5000. Owe $150.
3. Discover. Limit $1000. Owe $400
4. Bank Of America Signature. Limit $5000. Owe $600.


All of these balances will be payed in full. (I've never carried a balance)

My credit score right now is 760 according to a letter that I got from Bank of America and is 720 according to Credit Karma.

My question is a 2 part question.

A. What credit cards can I open,( I don't mind paying the annual fee is there is any) that would allow me to accumulate enough points via their offers, to be able to buy a ticket for her? London-NYC-London. I'm thinking Chase-freedom Business? Chase Sapphire?

I'd appreciate someone telling me which cards I'd need and in what order so that I could transfer the points between them to be able to buy the ticket.

B. If I cancelled these cards within a year, how many points would I stand to lose?


I'd appreciate any help! so a HUGE thank you in advance.

Also- If my first question is better suited for the "travel" thread, I'll take it there.. Sorry!
Just thought I'd ask! thanks!