How to prove OLS assumptions

Question

I'm writing my dissertation right now and in it I've used a regression model. With one independent variable and 4 dummy variables.

Y = b_0 + b_1STAR1 + b_2STAR2 + b_3STAR3 + b_3STAR4 + b_3 STAR5 + e

Were the beta parameters are OLS estimators and STARi are the dummy variables i.e( value 0 or 1).

Now i know what the 5 assumptions behind OLS. Although how do i prove that they are fulfilled?

1. Linear in parameters

How do i prove my beta parameters are linear?

2. Random Sampling

I understand how i can argue this assumption.

3. Zero Conditional mean

Can this be proven by a scatter plot?

4. Sample Variation

Whats the difference between Sample variation and Random Sampling?

5. Homoscedasticity

If I've done a regression in Excel can i just take the variance of all the residuals to see if its a constant?