The answers to your questions vary -- not because the answers are not precise but because Sharpe ratios are often used for marketing.
The Correct Estimator of a Sharpe Ratio
If we are trying to be correct (and not trying to fool ourselves), the expected Sharpe ratio $S_P$ for portfolio $P$ would be simply $E(S_P) = E\left(\frac{r_P-r_f}{\sigma_P}\right)$ for a risk-free rate $r_f$ (scaled for the time period). We often just plug in the estimated or average values yielding the realized estimator $\hat{S}_P = \frac{\bar{r}_P-\bar{r}_f}{\hat\sigma_P}$.
This is incorrect since, due to Jensen's Inequality, we should account for uncertainty (and skewness) in the estimate $\hat\sigma_P$. However, I'll leave that issue aside.
Annualizing a Sharpe Ratio
Sharpe ratios are like volatilities in that they are annualized unless stated otherwise. If you use daily returns, the $S_{P,\text{daily}}$ you calculate will be a daily Sharpe ratio and you would need to scale the mean daily returns and daily volatility for the number of trading days/year $N$ to get an annualized figure: $\bar{r}_P\times N, \hat\sigma_P\times\sqrt{N}$ or $S_P=S_{P,\text{daily}}\sqrt{N}$.
The number of returns used in calculating the mean does not matter for scaling; what matters for scaling is the period an average represents (daily, in your case).
To get a monthly Sharpe ratio, would would scale by the number of trading days per month; in your example, that would be $S_{P,\text{monthly}}=S_{P,\text{daily}}\sqrt{20}$. If you only have 7 days of data but could have traded the strategy for the entire month, then you should still take those 7 days as an estimator for the daily Sharpe ratio and get a monthly estimate by multiplying by $\sqrt{20}$.
Varying Dollar Amounts
If the 20 days had different dollar amounts, there are a number of options for computing the Sharpe ratio. The most correct approach would be to calculate the profits you made and express that as a return on the capital you used. Thus if you held some cash, you would (effectively) include that in the calculation. Slightly less correct is to ignore the cash and compute an internal rate of return -- and a weighted volatility. However, some people will also just cumulate the percentage returns as though no capital was infused or withdrawn during the period and compute the volatility on those unweighted returns.
This is where we start hitting the tension with marketing. The method used varies depending on which is most attractive (i.e. larger). Worse, marketing people may use the lowest of weighted or unweighted volatility with the highest of weighted or unweighted average returns.
Standard or Log-Returns?
That brings up another tension: though it is easier to compute a Sharpe ratio using log-returns, marketing people hate it because log-returns are slightly lower than standard returns. Standard returns (e.g. $(p_{t+1}-p_t)/p_t$) are preferred since they are slightly larger.
If you are computing a volatility, the returns averaged should be consistent with those used to compute the volatility. Guess what? Marketers do like volatilities computed from log-returns since those tend to be lower volatilities.
Other Skullduggery
We are not done with the truly reprehensible acts which may be committed in computing a Sharpe ratio. Some marketers will take the product of standard returns and then "average" by dividing the product by the number of days -- instead of doing the correct method of taking a geometric average. They may then divide that by a volatility computed from log-returns. Some will cherry-pick the time periods they get these figures from. They may also choose the minimal risk-free rate during the calculation period -- or the lower of a rate from the start or end of the period.
Finally, some strategies are "special situations" in that they exist for a short period of time and are not always open for investment. Examples include investing in distressed firms after bad news, earnings strategies, or index rebalances. If you cannot always hold a position in these strategies, some marketers will just scale the Sharpe ratio to an annualized figure -- even though the return and volatility you earn on an annual basis might be the same as that for the quarter in which the strategy was active.
If you are starting to feel unclean, remember this feeling every time you read a fund's claimed Sharpe ratio.
