You can test the assumption of the independence of the treatment and the covariate by performing an ANOVA on the covariate. In a randomized experiment in which the covariate has been measured before the treatments are administered, there is no reason to do this because the treatment cannot have an effect in this case. If the covariate is measured during or after the treatment, then the ANOVA on the covariate should be carried out because it provides information on the effect of the treatments.

For most nonrandomized studies, the ANOVA on the covariate should be carried regardless of whether treatments have been applied. If the treatments have not been applied but the ANOVA F is significant, then this is a warning that the ANCOVA F and adjusted means are almost certain to be biased as reflections of the treatment effects.

In terms of your last question: Yes, you can formulate an ANCOVA as a regression model. Specifically, if I had three groups I could formulate the ANCOVA model as:

Y_i = b0 + b1D1 + b2D2 +b3X_i + error_i

where D1 and D2 are dummy vectors and X_i is the covariate.