Question1: I think you're confused on what you're actually measuring. Don't think about this in terms of trades, think about it in terms of the total value of your portfolio. At day 1 you have 100 dollars, tomorrow you have 110, 2 days from now 115 and a week from now it's back to 105. How many trades you made during that period, how big those positions are or even how much cash you have is irrelevant. The time series of those portfolio values (100, 110, 115.... 105) is from where you calculate your standard deviation and average return to get your Sharpe Ratio.
Question 2: Even if you're working with daily returns and measure the daily standard deviation it's probably a good idea to transform those into yearly measures (in fact you need to do so in order to calculate your Sharpe ratio). Just take $1-(1+r)^{252}$, where r is your average daily return calculated in question 1, to get your average yearly return and $\sigma_ {daily} * \sqrt{252}$ to get your volatlity as a yearly measure. The interest rates you found are already measured yearly. the correct rate to use is the theoretical risk free rate, so none of them really, but the 10-year rate should be fine for your purposes.
Question 3: Again, not sure what you mean. Do you mean if you should use the geometric or arithmetic mean when finding your average rate of return? I would use the geometric but I'm sure you'll be able to find people with differing opinions.