Learn English – How should I ask a polite question to the manager

Both uses of the past form are possible. Some grammarians speak of the past form as "remote", because it "removes" the verb to a distance which may be either temporal or social.

Hi, Maria, I'm calling because I wanted to take you out for dinner.

Here you use the past form to make want less demanding. The remoteness is social.

Sorry I couldn't track you down last night. I wanted to take you out to dinner.

Here the primary sense is clearly temporal remoteness. You could, however, work in a degree of social remoteness by also employing the progressive construction, which in this case would have no implication of imperfectivity - it would just be a further distancing device.

Sorry I couldn't track you down last night. I was wanting to take you out to dinner.

Note that this last construction could also be used in the first example:

Hi, Maria, I'm calling because I was wanting to take you out for dinner.

In this particular instance, however, the progressive construction would probably not be used because it clashes with the progressive calling in the main clause.

• Grammarians give this the Latin name horror aequi, "the widespread (and presumably universal) tendency to avoid the use of formally (near-)identical and (near-)adjacent grammatical elements or structures" (Rohdenburg).

You'd be more likely to use it in a context without a prior progressive:

Are you by any chance free tonight? I was wanting to take you out for dinner.

To express the "see-what-you're-missing" sense in English you would employ a construction with a stronger sense of intention. The past would be necessary in this case, rather than optional, because you would be speaking of a prior intention which no longer holds:

I was going to take you out for dinner, but I've changed my mind.

In general, it is more polite to ask for help achieving "something" than to ask directly for the "something".

I would like to ask you if it is possible to secure me an internship with a software company.

could be rephrased as

Would you be able to help me secure an internship with a software company?

First, we turn our thought into a question so that it seems less like a demand, and then we ask if it is possible that she might be able to help. This way even if she can't directly find you a position, she may still be able to help by introducing you to someone who can.

I would also rephrase the second and third sentences:

I will have approximately two weeks free between first and second semester, and I would like to use that time to increase my experience.

This isn't English advice, but I can't help but add that in my experience in the US, two weeks is a very short time for an internship. Instead of asking for help getting an internship, you may want to ask for her advice on how to get practical experience during that time.

I was a part time intern for a professor in the Engineering school while I was pursuing my degree, and it worked out well because my work was close to my classes and the university had a process in place for hiring students. A position with an outside company is more difficult to secure because there is a lot more cost in hiring a new intern and they will likely want a longer commitment.