Why talk about entry-level, and assistance? Let them decide the caliber of jobs they'll give you. There's no reason to request those positions. Just list your skills.
Remember that this company isn't actively seeking someone for the role you want to fill. If your letter is too formal, they'll just say "Were not hiring right now" without giving you an extra thought. You're looking for a summer position which only lasts 3 months. Your email needs to show that the cost of hiring you, which includes training and your pay, will be less than the amount they'll earn from you. Why won't it cost a lot for them to train you?
Also, never "put all your eggs in one basket." I find that it's a lot easier to get someone to like you if you have a few emails back and forth. If you give them everything, they'll have no reason to email you back.
Lastly, be open about compensation. Don't let them get away with paying you less than the cost of lunch and transportation, because this certainly shouldn't cost you anything. However, realize that this company is doing you a favor, and you shouldn't be expecting to get paid anything near what their long-term developers get paid. If you look at this as more of a learning experience than a contracting job, chances are food and transportation is enough of a stipend.
I'm speaking from my own experience, so no promises. I did this two years ago, and tried a lot cover letters for a lot of different companies. Ultimately, the one below is what worked out for me. There were no attachments to this email. I received an email interview a few days later, then a small code test, then went in for an in-house interview.
My name is [name] and I have extensive experience with PHP, MySQL, and AJAX technologies. I have developed a variety of applications ranging basic form submission to live multiplayer AJAX games.
Currently, I am seeking an internship for part of my senior year curriculum. The internship will be required to meet [my internship details]. I have worked freelance for the past two years and would love to secure a steady stream of work. Compensation
can be discussed.
Please let me know if you are interested and I will send more information. If you would like to see some of my work samples do not hesitate to ask.
The best part about this email is that the "senior year curriculum" I was referring to was for high school. They didn't figure out I wasn't in college until the in-house interview. I don't think you'll have to worry about this.