Well, there's a good reason they didn't destroy Jerusalem or Mecca... They would need to get to Jerusalem or Mecca first. The Mongol Ilkhan Ghazan Khan may have controlled Jerusalem for, at most, four months in 1300 before retreating out of Syria entirely (there is much debate if they took the city or not), but by then the Ilkhanids were in the process of converting to Islam and probably wouldn't have wanted to destroy the city as a result.

Even when the Mongols first entered Palestine in 1260, the leading general Kitbuqa was a Nestorian Christian and the Mongols had begun making negotiations with the Crusaders over the status of Jerusalem. There was no interest in just destroying it. (Besides, it had just been destroyed in 1244 by a group of Khwarzemian mercenaries hired by the Ayyubids.)

The bigger answer to your question is that Jerusalem and Mecca and are not strategically important for the Mongols. Mecca is way out in the middle of nowhere. Controlling it would give a Muslim ruler some added prestige, but otherwise...

Jerusalem had also lost a lot of its strategic value thanks to the Crusades. The city walls were regularly torn down (to let anyone know if they take the city, it will be taken right back), for example.

When Jerusalem was in reach of the Mongols, something much more important would be happening in another part of the empire that would require attention. In 1260, Hulagu had to go to Mongolia to deal with the succession after the Great Khan Mongke died, taking enough of the army with him that the Mamluks were able to defeat them at Ayn Jalut.

When Ghazan and his general Mulay raided Palestine in 1300, it was following a surprise victory at Aleppo and they primarily chased the fleeing Mamluks back to Egypt. After a couple of months, they decided to retreat back across the Euphrates, probably because they needed better pastures for their horses.

As for the first part of your question, what is the religion of the Mongols? Well, that's a bit of a vague question. There were lots of different kinds of Mongols at different times. During the life of Genghis Khan, they were mostly Shamanistic. The Ilkhanate successor state (the Mongols who ruled Iran and Iraq) were Christian for a period of time and then converted to Islam. The Golden Horde (the Mongols who ruled Russia) were actually the first Mongols to convert to Islam. In the east, Mongols took on different forms of Buddhism. In a lot of ways the Mongols were fairly tolerant of different religions, maybe even curious about a variety of religions.

Don't have my sources for anything specific here in front of me, but any survey of Crusading and Mongol history should get you to them. I would recommend Riley-Smith's The Crusades and David Morgan's The Mongols.

There's a great book on the Mongols by Jack Weatherford. You can find it on Googlebooks, although you won't be able to view the entire thing you can see quite a few good bits. You can find it here, its well worth the read even if some of his ideas seem a bit grandiose. All of the following comes directly from that book.

The religion of the Mongols was essentially shamanism. As their empire rapidy expanded they obviously encountered many diverse peoples and religions, and many of the Mongols converted or caused others to convert as pretty much all of Central Asia was in flux during and after their expansion. As for how Genghis Khan saw it himself:

Genghis Khan recognized the disruptive potential of competing religions. 
In one form or another, virtually every religion from Buddhism to 
Christianity had found converts among the steppe people, and almost all 
of them claimed not only to be the true religion but the only one. In 
probably the first law of its kind anywhere in the world, Genghis Khan 
decreed complete and total religious freedom for everyone. Although he 
continued to worship the spirits of his homeland, he did not permit them 
to be used a national cult. (pg. 69)

There's also a very interesting tale about a great religious debate hosted by Monke Khan that took place in the capital of the empire. Between Christians, Muslims and Buddhists, it basically ended with every one of the debaters passing out drunk after no side was able to win the debate in anyway. Provided its even half true, it shows you the very open attitude that the empire took towards religion, at least officially. That section is on about page 170-173. I wish I had the time to go through the book and find more interesting tidbits, but its been awhile since I've read it and cannot recall any off the top of my head. I'd recommend looking through it yourself though, you can easily find relevant sections with the search function on Googlebooks.

Unfortunately I don't think I can sufficiently answer the second part of your question, although in the case of Mecca at least I feel it would have to do with logisitcs. Even if they wanted to destroy Mecca, it would have been very difficult or rather impossible to support all of their horses in a region of mostly desert with few places for grazing, and the Mongols would have brought tens of thousands of horses with them.

The religion of the Mongols was Shamanism (at least at the beginning). "http://en.wikipedia.org/wiki/Religion_in_the_Mongol_Empire"
They did not believe that it mattered outside of Mongolia and so there was religious freedom in the empire "Chua, Amy (2007). Day of Empire: How Hyperpowers Rise to Global Dominance—and Why They Fall. New York: Doubleday. ISBN 978-0-385-51284-8".
I am not sure if that had anything to do with not destroying Mecca or Jerusalem.
Sorry for citing Wikipedia. I don't have any of my other books with me as I am in Mongolia.
Edit: formatting
Edit: Good answer as to why Mongolians were tolerant of other religions here: http://www.reddit.com/r/AskHistorians/comments/18d8ed/what_was_ghengis_khans_motivation_behind_his/ Credit to: /u/alltorndown

Look up Harold Lamb's book on Ghangus Kahn. He is pretty in-depth in his coverage. Lamb does state that Ghangus traveled with a entourage of scribes, priests, and scholars, as so he could learn about the different peoples, and also had them teach his family. Ghangus was of a traditional Shamanistic faith that was natural for Mongolia. But it is questioned if he was still when he died. Timmigen, is actually a Fascinating person to read about, and had a lot of things most people did not think of in his time, in many ways he was actually ahead of his time, just did not have the training to do what he thought of. But later learned how to, and then there was a whole new world from there on.