Is this a cultural difference between continental European and British/American system
I cannot surely speak for all continental Europe, not even for a small portion of it and not for all fields, but speaking with a few European colleagues of mine, it appears that the expectations that we have on recommendation letters are completely different from the typical US expectations.
- We mostly consider recommendation letters as a mean to discriminate between those that are not able to provide any recommendation letter and those that can provide at least a couple. That is, we expect most of the people not to be able to get one (in a distant past, I wouldn't have considered one who just attended one or two classes with me eligible for a recommendation letter).
- We don't expect recommendation letters to be more than one page long, and half a page is ok (you just have to tell me that you had worked with this person and that they are not that bad).
- We don't expect the recommendation letters to be sent directly from the reference to the application board.
- We consider the writing of recommendation letters as a favour to the applicant, and not as a duty of our profession.
Actually, all the recommendation letters I wrote but one were handed directly to the student (or former student). For the exception, I've received a direct request from a university to submit the recommendation through their online system: the uploading procedure, with questions to answer too, was such a hassle that I've decided that in the future I will probably decline any request for recommendation letters to be uploaded anywhere.
To answer your main question:
Should I ask the professors to write and hand me recommendation letters directly, or should I try to communicate with the university and let the professors email them the letter/give me a physical letter sealed within an envelop, which I will send to the university by mail later on?
I understand your reasons, but I'd consider this an odd thing to do. I'd simply ask the student to provide the recommendation letters.
In case the professors were reluctant to hand the letters directly to the student, I'd write an email explaining the different traditions.