They aren't under half strength. They're at exactly half strength and get their full leadership value.
As far as I know, you CAN move back into position after regrouping in a LMS situation (with marines)I'm not at all sure you can move after you regroup in this situation.
Normally you can do what you like if you fall back due to enemy action in your opponent's turn. You regroup in their turn at the end of your fall back move and begin your turn not broken, so you can act normally.
However in this case you are falling back and regrouping in your own turn. As a result you end up being able to do the same kind of things that any other unit which regrouped in its turn. As I understand it you get a 3" consolidate type move, you can fire (counting as having moved I think, since you did) and you can assault, but you don't get a standard move or turbo boost.
That makes sense to me. You would be moving twice in the turn otherwise. It doesn't seem right at all that a unit could fall back 3d6" and then turbo boost 24". It would have to go very fast for that to work.