To get the marginal distribution of just one rv, you need to sum over the other one.

For example P(Y=0)=P(Y=0 and X=0) + P(Y=0 and X=1)=.4

You are summing over all values of x

P(Y=1)=P(Y=1 and X=0) + P(Y=1 and X=1)=.5

Since these probabilities sum to one

we have .1 left over for P(Y=2)

The last probability, Y=3 & X=1 =>0 should be erased.

Anything that has probably zero shouldn't be listed.

From here you should be able to get Y's mean and variance.

As for the conditional mean of Y when X=1,

you need the distribution of Y when X=1.

You only need these two

Y=0 & X=1 =>0.2

Y=1 & X=1 =>0.3

P(Y=0|X=1)=P(Y=0and X=1)/P(X=1)=.2/.5=.4

Thus P(Y=1|X=1)=1-P(Y=0|X=1)=.6

So, E(Y|X=1)=(0)(.4)+(1)(.6)=.6

The correlation between two rvs is the covariance divided by the st deviations

AND by the Cauchy-Schwarz inequality it has to be between -1 and 1.