Why are you required to pray at certain times during the day facing Mecca?
It is a pillar of Islam to pray 5 times a day (at dawn, at noon, at afternoon, at dusk and in the evening) and we all pray facing one direction, which happens to be the ka'aba in Mecca (which is also the place where Muslims go for pilgrimage). As for why, the answer to this question is the same as the answer to any "whys" about Islam: because Allah (God) commanded us to.
Why do you not pray towards heaven?
Muslim prayer consists of a series of bows and prostrations... you cannot really prostrate towards heaven since heaven is above everything. Here is a video of Muslim prayer, just so you can get an idea of what I mean:
Are Muslim prayers all set prayers.
No. This is the "formal prayer" if you will... You could always just say "Dear God....." without the actions, and ask for what you want.
Is the wording the same for them?
Yes, but you can also add on to it if you want.
What do you do if you are at work/school and it is time to pray and there is not place to pray in the manner shown?
Prayer is very important to us, and you cannot miss prayer. It has to be done during its "time slot." Personally, if I am in school, I find a quiet corner, or pray near a deserted staircase. In the mall, I have prayed in a changing room before. When traveling, you have a little bit of leeway (you can for instance, pray the prayer shortened, and pray two prayers at once). The place for prayer just has to be clean and no one should pass in front of you. This can be done by taking a clean rug with you and laying it on the floor (or sometimes when I am outside, I just put my jacket on the floor) and if you put an object in front of you, then no one would be passing in front of you, or you could pray facing a wall.
Is personal prayer part of Muslim prayer as well or is it more formal prayers with set words and set times of the day?
Yes, personal prayers are also done. They can be "informal" (just talking to God), or you can pray extra "formal" prayers, or you can add your own words to your daily 5 prayers. The "informal" prayers are called duaa, which is translated as supplication, and there is a duaa for pretty much everything under the sun (trying to remember something, wishing someone well when they are sick, congratulating someone on marriage, etc, etc). Also, you could always just make up your own prayer (supplication) and ask God for whatever you need.