Regression Trees - Splitting and decision rules
Short Answer

1.Yes feature variables are needed for split. No the best 2 subsets for mse reduction are not created. 
2.Yes

Long answer
Decision trees are greedy algorithms that choose a feature and split at each node and use that feature and cut to divide the data.

So as you mentioned in point 2 the tree starts building with all y values in the first node and iterates over all combinations for each feature and selects the best feature and value split to divide into 2 groups which are further split.

The termination conditions would be number of observations in the leaf node and/or the threshold for decrease in mse from in the split.