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What is the effect of increasing the sample variance?

As a sample size increases, sample variance (variation between observations) increases but the variance of the sample mean (standard error) decreases and hence precision increases.

When all other factors are held constant What effect does increasing the sample size n have on a confidence interval for a mean?

If other factors are held constant, increasing the sample size from n = 10 to n = 20 will increase the width of a confidence interval.

Does increasing sample variance increase likelihood of rejecting null hypothesis?

When we increase the sample size, decrease the standard error, or increase the difference between the sample statistic and hypothesized parameter, the p value decreases, thus making it more likely that we reject the null hypothesis.

Which of the following would happen if you increased the sample size?

Increasing the sample size decreases the width of confidence intervals, because it decreases the standard error. This can also be phrased as increasing the sample size will increase the precision of the confidence interval.