Native English speaker answer. I have been a University lecturer (in Science, not in English, but with many non-English undergraduate and postgraduate students). A really common mistake that non-native speakers make is the original poster's Case 1.
Sentences of the form "It allows to do something" should not be used. Here's a typical sample that I found in a recent written experimental report from a student who, nevertheless, had excellent oral communication skills in English:
"As this system only allows to image samples with
length and diameter smaller than 2.5cm, due to its reduced field-of-view, a new system was required."
This needs to be rephrased as either:
"As this system only allows me/one to image samples smaller than 2.5 cm..." (Case 3 above)
"As this system only allows imaging of samples smaller than 2.5 cm ..." (Case 5)
"As this system only allows samples smaller than 2.5 cm to be imaged ..." (Case 6)
(BTW For the purists out there, all these sentences contain another, more subtle, error. Here what is allowed is the imaging only of samples smaller than 2.5 cm. It's not the case that the system "only allows". It might also do other things as well as "allow". E.g. "The system both allows and encourages the imaging of such objects." Thus, the "only" is technically in the wrong place in the original sentence. However, this imprecision has become so ingrained in native speech that it is now accepted usage.)