Short answer: Yes, they teach foreigners. No, it wouldn't be possible for an outsider to become a monk there. You can, however become a disciple under a monk. And you can become a sifu of your own school while being a disciple of a monk.
At the Shaolin Temple at Songshan in Dengfeng, Henan, monks routinely take on foreigners for training. Usually the junior monks are assigned to teach outsiders. Your training there will be whatever you ask for. You pay for it, and they provide the training. The senior monks will figure out what you want and assign the right junior monks to teach you. You decide how long you want to stay. You can be there a week or two, or you can be there for years. It's up to you and your wallet.
You will not stay at the temple. You will sleep and eat somewhere else. The monk in charge of you will often set this up for you. They'll find a place for you to stay (usually at the nearby hotel), and they'll provide all your meals. They'll provide transportation to take you to Dengfeng to buy your monk's uniform, leggings, weapons, and so on. They'll even arrange for tour guides and translators if you need them.
A typical day begins at 5:30AM with a run up and down the mountain. Then you meet with a monk for training. Then you eat breakfast wherever you want (not at the temple). Then you come back after breakfast for more training at the temple. Then you eat lunch. Then you come back for more training at the temple. Then you eat dinner. And finally, you go to sleep.
This is very hard on your body. It will take weeks to become accustomed to the altitude, the training regimen, and the food. Expect to have diarrhea your entire first week, by the way.
You will not get trained the same way monks would be trained. The monks there start at an early age. They are hand selected from the best of the best. They undergo a rigorous, structured training regimen from an early age and live there at the temple. They don't get any say in what they train in. They have to learn to read, write, and do math. They have this academic education in addition to their wushu training. And they have to learn the Buddhist rituals.
From what I gather, the religiosity of the Shaolin monks varies. A small number are very interested in the Buddhist religious aspects. Others don't care at all. Most probably care a little. With some, they only become really interested in it later on in life, in their 20's and 30's.
Most of the monks are there to train in wushu. That's their dream. They want to be the best at it. They often have the goal of leaving the temple to become either businessmen or movie actors. They're not usually interested in the religious aspects.
For business, many of the monks that are currently living at the temple also have connections with martial arts schools in China and elsewhere. The teachers of those schools pay them to be the head instructor of their schools. It gives their school credibility and status having a real Shaolin monk as their head instructor. The monks rarely visit the schools, if ever.
There is a rule that the monks must live at the temple. They can't just go off and have an apartment and set up businesses, etc. So they have cell phones and computers to connect them with the outside world. They receive money electronically.
Now, as for you wanting to become a monk, it's not possible. But you can be a disciple. You just need to find a monk there that is taking foreign students and ask to become a disciple. After taking the oath and performing the ritual (I've witnessed this myself), you will receive a Chinese monk name. But you are not a real Shaolin monk, just a disciple. They probably have a special term for this, I forget.
Eventually, your monk will award you with sifu title if you've been with him for long enough and have become suitably skilled. And then you can open a school and start teaching. You can still open your own school and begin teaching before that, but you just can't refer to yourself as a sifu before your monk grants you that title.
I believe your monk will also award you the title of master. I'm not sure.
I asked a friend of mine who happens to be a disciple under a Shaolin monk and who is a sifu what the oath of discipleship involves. And specifically if it involves becoming a Buddhist in any way. Here's his answer:
The discipleship oath consists of 5 core vows of which being Buddhist
is not one of them. The monks make 7 vows, 2 more in addition to their
core discipleship vows which include taking refuge. In a nutshell and
not in any particular order, the vows are:
- Being a good person and compassionate and helpful to those in need.
- Defending Shaolin Temple.
- Not killing.
- Refraining from vices, i.e. smoking, sex, alcohol.
- Loyalty, Honoring Shaolin Temple and your master.
And I go over my experience with training at Shaolin a little more at my answer here:
Training martial arts in china
Hope that helps!