I believe my foreign sounding name makes it more difficult to get a job as a software engineer. Are there any statistics backing my observations? And should I put an Anglo/American name on the top of my resume?
No. If you get past the interview and they do a background check and the names don't match, that could be perceived as misrepresenting who you are - or worse - as lying - unless it was very obvious. That could be quite fatal.
That being said, you could include an Anglicized nickname that would indicate you prefer to use that name.
That will make your resume stand out, as well as communicate the name you would prefer them to use, all without having to take the risk of being perceived as misrepresenting your name.
For example: Dylan "Jack" Chensky
A resume is a marketing document, not a binding legal one. It's best to use whatever name you go by on your resume, as opposed to your full legal name (unless your name is Benjamin Franklin du Pont or Maddox Chivan Jolie-Pitt). This also goes for LinkedIn, your professional website, and any other professional/marketing documents you have out there.
There is ample research on the bias at play when someone reads your resume. You're more likely to get an interview in a western country with an Anglo-western name. Regardless of how difficult it is to pronounce your name, picking an anglicized name to go by is a strategic move. Why wouldn't you give yourself every advantage, especially given that finding a job can be hard enough?
Personally, I use a shortened version of my name on my resume. Not because I have a foreign or difficult to pronounce name, but because my full legal name is quite long (around 30 characters) and it looks awkward on the page. Also, nobody calls me Elizabeth (my legal first name) and I would prefer it stay that way.
To answer your question: Yes you should absolutely put an anglicized name on your resume if you're applying for jobs in the US.
I personally find it a little patronising when people pick a "fake" anglicised name because they think I won't be able to pronounce their real name - I would recommend you simply use part of your real name that is short enough to pronounce.
Its quite normal to shorten your "first" name. For example, if your name was Jonathan you could use Jon. This is not considered dishonest or secretive as everybody knows that many first names are short forms of longer ones.
If that doesn't apply (depending on culture), you can also use your given name or equivilant, e.g if you were Li Min Chen where Chen is your given name, you can go by Chen. In this example, its one syllable, nobody should have any problem.
You can use your full name on anything official.
The US is such a diverse country with people from all over ending up here, I don't think I would worry about it too much. I've seen resumes with all kinds of complicated names on them.
I don't think I would put an alternative name on the resume itself, but offer it in the signature of a cover letter or at the initial contact for an interview. This is similar to someone who might go by their middle name, they would usually put their official name on the resume and then offer another one on contact.
Just don't use Kanji or Cyrillic, give them a shot at pronouncing it.