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Similar as RRSP, you can open another or more accounts with other banks at any time, next year or this year, as long as your total contribution among all accounts does not exceed the amount allowed by CRA.  It is the banks who report your last year TFSAs' activity to CRA and CRA then tells you how much you can contribute for this year, based on last year's reports and current year's amount, it is in your tax Notice of Assessment from CRA.  

Say you contributed $5000 and then took out $2000 from TFSA in 2009, CRA will tell you in your 2009 notice of assessment that you can contribute $7000 in 2010, and it doesn't matter if you contribute $700 each to 10 accounts or $7000 to one account in 2010.  

Of course people normally like to consolidate their accounts for better investment management.  So if you like the other bank (say BMO for example ) better, you can also transfer your TFSA money from HSBC to BMO, just like RRSP transfer (but with a different form), with no impact to your new TFSA contribution room.

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(1) go to the bank where you want to transfer TFSA to, they will help you to fill the form and send the request in, similar as in RRSP transfer case.  Remeber to ask they cover the trasnfer fee (charged by the bank that tansfer from) for you as well.

(2) In theory yes, but first, no banks are not open on Jan 1st; second, why do you need the take the money out for only one day in a TFSA account?  To be safe, you'd better wait the notice from CRA before contributing, or call them up to ask.

The benefit of a TFSA is for long term investment, not good for normal banking usage, as Lina also mentioned.  It can be a great thing only if you use it correctly.