It can be either her or his. In the past, people tended to use masculine pronouns more. But as society has become more equal between the sexes, you will see use of both masculine and feminine pronouns almost equally. It's really up to the speaker which they use. I think "their" can also be used here, but I've always found it awkward. It is used in order to ignore gender, but at the same time it ignores the plural. Good teachers have to be able to motivate their students. That would be a good use of "their".

Perhaps because the author is female. Perhaps because there are more female teachers than male teachers and the author chooses to speak to the majority. It's a personal choice. The author could have said "be able to motivate his or her students" as well.

All three are grammatically correct. Really, it is up to the author to choose. Choosing to use 'his' or 'her' just adds a bit more information to the context, whereas 'their' is less informative.

It should be "their". But the speaker was thinking of female teachers at the time.

You could use 'his' instead of 'her'. Most teachers are female (her) so that is most common to use, but for purposes of correct grammar, you could say "his" instead but you could not say 'their'. 'Their' is plural (more than one). "A good teacher.." is singular (only one).