Your significance level for hypothesis testing is denoted by alpha, it is, basically, the threshold that is assigned by which chance is a reasonable explanation between the variation between the data and the null hypothesis. Meaning, if you pick an alpha of .05, which is a common level, you are saying that if the the chance of the data you have occurring assuming the null hypothesis is any less than 5% I will reject the null hypothesis.
I'm not sure what you mean by level of confidence, but I'm assuming you're talking about the confidence interval, which is directly related to your choice of alpha (the larger your alpha level, the smaller your confidence interval. After all, if you need 100% of your evidence to conform to the null hypothesis regarding a mean, you are 100% confidence in that mean, and therefore there is no interval) but also is related to error.
About your two questions, I'm a little confused. By estimated mean, do you mean the mean you are hypothesising that the true mean is, or do you mean the sample mean? When you are building a confidence interval of the mean then it is centred around the sample mean, and it not guaranteed to include the actual population mean, and it won't sometimes because of chance and type 1 error (rejecting the null hypothesis when it's true, which you will do 1/20 times with a .05 alpha level)