Can I apply for a Schengen visa for an earlier period than my existing one?

In your case it is now academic (though reporting back your experience would be appreciated) but I am not aware of any regulation to prevent what you suggest and see no reason for one. Schengen visas can be requested up to three month in advance of planned travel and clearly in those three months circumstances may change and give rise to a need for another visit to the Schengen area.

Note also that processing time is likely between 10 and 15 days, possibly more, and your non-routine situation might mean longer than usual. For certain nationalities at certain consulates the processing time may however be much less, "usually 3 - 4 days" is mentioned here but the same site also mentions: a MINIMUM of 15 days.

Nevertheless you should still have time to fit in a quick visit sometime between about May 5 and May 18. Since you would have two visas (the existing multiple entry and, hopefully, a single-entry one) you would need to leave the Schengen area before May 18, even if only to reenter on May 18. If granted a single-entry visa it may well be for a duration of only a few days.

However long you spend on a single-entry visa, the 90/180 day rule applies regardless of the number of visas (and of the number of passports). Rijksoverheid states:

If the visitor used up all the 90 days, he/she has to leave the Schengen area and may only reenter after 3 months of being abroad.

but this is not a requirement for 3 months between visits, rather that after visiting for 90 days a further 90 days must be spent outside the area to comply with the 90/180 rule. Assuming no other visits are involved, a 5 day visit in May under the single entry visa would leave 85 consecutive days available under the multiple entry visa from commencement later the same month.

There is a calculator for the 90/180 rule here.

Considering that you have a student Schengen visa, you cannot travel to the relevant Schengen country before the visa validity. Once a Schengen visa has been issued it must either be used or canceled. There are no "date change" options on it like a travel itinerary.

Futhermore you cannot have two valid visas for an overlapping duration for almost any country.

What you can do is either:

1. Get your current visa canceled and apply for a new one with the proper dates
2. Approach the embassy which issued the visa and ask the visa counselor for the best course of action (I'm pretty sure he/she would also cancel your current visa and issue a new one)