The payment of interest is straightforward. The central bank is a bank, and pays interests on deposits in exactly the same way as any other bank.
If the central bank wants to pay $1 in interest to a depositor, it:
- increases the amount on deposit by \$1 and
- reduces the central bank's equity by $1.
That is, it just reshuffles the right hand side of the central bank's balance sheet; the asset side is unchanged. (On the income statement, there is a correpsonding $1 interest expense.)
All the central bank needs to do is ensure that the aggregate interest paid on reserves is less than the aggregate interest received from its lending and bond/bill holdings to prevent equity from going negative.
Since 100% of a central bank's assets pay interest, and its currency (dollar bills) do not, keeping the net interest flow positive is not too difficult (in a positive interest rate environment).
You asked what backs the currency. The answer is the taxing power of government. There's a few theoretical justifications for that statement, One is provided by Fiscal Theory of the Price Level (edge of mainstream), and another is referred to as "neo-Chartalism" (post-Keynesian).
As for the relationship between inflation and the rate of interest, the answer depends upon which school of economics you follow. Increasing interest rates increases the interest income of the private sector (which should increase effective demand, as you note), which is traded off by the effect on behaviour of the higher rate of interest. The conventional answer is that the behavioural effects dominate; that is, higher interest rates slow the economy and inflation.