Sure, I can answer some of those questions. I'll start off by saying that you're right that the process will be more difficult as a non-native English speaker. But, after you've sent the first few email, you should be able to see patterns between them, and you'll end up with a standard opening email and a standard set of responses. Also, it helps to have a friend glance over the email. I'm a native speaker and I still used my friends to occasionally proof read my work.
As for who to contact in the company, I would say that you try to contact the person who would be your direct supervisor in your new position. They're going to be the person who is best able to evaluate whether you are a good fit for the job, and if they decide they want to hire you, they'll make it much easier to go through all the paperwork. If you don't know who your supervisor would be and it's a large company, try to find a senior engineer at the company. If it's a small company you could contact the CEO. These people at least will have a decent idea of what positions are available and who you'll need to contact to talk about them. Avoid HR at all costs. Also, since most "careers@companyname.com" email addresses go straight to HR, try to avoid those as well. Finding the email addresses can sometimes be tricky, but I was usually able to find something by googling peoples names/positions/companies.
When I say "make it as easy as possible for them to get back to me" it's less that I'm trying to write a simple letter, and more that I'm only asking them for a small thing. Asking someone for a job is a large thing to ask for. Even asking for an interview is a large time commitment. You simply want to ask them for information about the positions they have available. After you've received that first email it is much easier to turn around and propose a time to talk on the phone.
I'll give you an example, from start to finish of what this conversation could look like. I've written the first email such that you could fill in the blanks and send it off. I've structured it such that the person reading could read the first paragraph and immediately respond (the phrase "what follows is a brief description of a recent project" tells them that if they want to skip that paragraph and just email you back they can). The project is probably the best way to do three sentences about yourself, but if you wanted to change it to highlight more soft skills you could.
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First email to a company:
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Subject: What's happening at <Company Name>?
Hi <Contact Name>,
My name is Hyunwoona and I'm a Computer Science student at <College Name>, with a focus on <Technical Skill relevant to the job>. I've been looking for internships, and came across <company name>. I was particularly interested in <specific thing the company is doing>. I'm hoping you could tell me more about the work happening at <Company name> and about the positions you have available.
What follows is a brief description of one of my recent projects:
As a project for <company || open source project || personal reason || class name> I worked on creating a <what you crated> using <language>. My focus with this project was to <description of why the project was worth doing (like: it's never been done before, needed do it faster, or make it clearer)>. I particularly enjoyed getting to see <specific technical aspect of the project> up and running. <If possible, add a sentence with a link to the code>.
Best,
Hyunwoona
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Response from company:
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Hi Hyunwoona,
The work we're doing at <Company Name> involves a lot of <stuff> and <other stuff> right now we're working on <some broad vision>. We are looking to hire interns in the coming <time period>
We're currently looking for interns for the following positions:
1) <Job Title A> -- This position involves <Something related to what you're capable of doing>
2) <Job Title B> -- This position involves <Something unrelated to what you're capable of doing>
If either of these are something you're interested in, we should discuss further.
Best,
<Contact Name>
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Follow Up email:
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Hi <Contact Name>,
Thanks for the response. I think I would be a good fit for the <Job Title A> position and I'd love to talk more about it. Can we set up a time for a phone call? I'm available <times when you're available>.
Best,
Hyunwoona
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Of the emails that I sent that led to interviews, more than 2/3 went almost exactly like that. Occasionally, I would have sent it to someone who would pass my email along to a colleague, but from that point forward the response I would get would still be pretty similar. I hope this helps. Let me know if I was unclear about anything.