It doesn't make sense to list a start date before the resume submission deadline. I would tend to suspect that there is a typo in the posting. Normally, if a company is posting positions on a university's career center web site, the start date would correlate to the end of the semester-- posting a position that is supposed to start just when the current semester is winding down would be quite odd. My best guess is that the company intended the start date to be August 15 which gives them time to interview candidates and probably coincides with the end of the summer term. Either that or the submission deadline is April 15 and the start date is July 31.
I would talk with someone at the career center, though, to try to understand the odd dates. It's possible that some dates were transposed in the process of posting to ad.
As a general rule, you probably don't want to reach out to the company (other than submitting your resume, of course). More often than not, the hiring manager is rather busy and the company has a relatively standardized approach to hiring. The company is probably also getting a large number of resumes for a given position. If you call and try to have a conversation with the hiring manager particularly before they've expressed an interest in interviewing you, you'll probably interrupt them doing something they consider more important and it is more likely that you'll come across as needy and annoying than enthusiastic. You're generally better off letting them review your resume and decide if they want to interview you before you start asking questions.
Now, occasionally, a really enthusiastic candidate cold calling going above and beyond does make a really good impression on the hiring manager and gets the job. That is the exception, though, not the rule. It's more likely to work for certain types of jobs-- a sales position that involves cold calling, for example, where persistence and an extroverted personality are important attributes of a successful candidate. It sounds like you are probably looking at some sort of software development position. That's generally not the sort of position where it makes a lot of sense to cold call the hiring manager. Plus, if you're asking the question, it's likely that you don't have the personality to pull off a successful cold call (nothing wrong with that, it just takes a very particular sort of outgoing and boisterous person to really pull it off).