If a Sentinel can be shot at any of its base squares, can it also shoot from any of them?

A Sentinel only has one white dot in the middle of its base. Double base characters have a white dot on each of its base squares.

Yup. Same thing with movement: pick a square and count from there... then when you get where your're going, only one square of the base needs to actually stop in the final square, allowing you to 'swing' the base around and get an extra square of movement if you wish.

Can a sentinel be merc'ed to X-men TA and get clicks of life back with the TA?

Also if Sentinel is merc'ed to batman would you be able to usethe Deep Shadows feat to stop any fire from opposing figs? If so capture has never been so good.

I don't see why not. The Sentinel just needs to be adjacent to a higher pointed X-Man, but he could even use the TA to trade clicks with a lower-pointed X-Man as long as that higher-pointed X-Man is there.

As to your second question, yes. Deep Shadows makes all terrain hindering terrain. So like I said before:

Sentinel + Mercenary + V Batman + Deep Shadows FTW

Keep in mind a Sentinel cannot be healed abov his starting click though - in other words, you cannot use the X-men TA to make a 100 point Sentinel into a 200 point Sentinel.

Yes yes, that is correct.