When The Doctor regenerate, he turn up with a completely different body.
All can be different from his previous regeneration, different hair color (''I wanna be ginger, I've never bin ginger before! --The Tenth Doctor), different face (''Imagine me with no nose!''--The Ninth Doctor), different legs (''I still got legs! Good!''--The Eleventh Doctor). Even a different character.
The things that don't change is his memory and his good will to save civilisations.
No, he doesn't stay the same height mainly because that would involve finding actors that are all exactly the same height. His whole appearance changes, so height is included, some of his regenerations are really tall, the fourth doctor being the tallest, the shortest one's are the seventh and the second doctors.
They cast actors which are of different heights, so obviously his height must vary too. Why would everything but height appearance-wise change but height stay constant?
