You see that especially in April and September when temperatures tend to be much cooler then most of the summer, alltho pitchers have been trained to baby their pitching arm, so you see many pitchers do that through out the year.

Most of the time it is personal preference or just comes along with the uniform. A lot of these undershirts help to take in the sweat so the uniform doesn't get muddy or is uncomfortable.

Wearing just the jersey alone is not all that comfortable but wearing a loose spandex undershirt is very comfortable

Because they, like all other people, sweat and want an under shirt to soak it up. Maybe because they have a ritual or a preference. Also, if its cold they wear sleeves.