I wouldn’t say it’s a “rule” per se, it’s just how most people pronounce it.

Granted, pronouncing the “u” in がくせい might make you sound a smidge “odd” to native speakers, perhaps, but it wouldn’t be because it was incorrect, just unusual.

I wouldn’t sweat it for now. In time, as you study further and listen to native speakers talk, you’ll start to pick up the flow on your own without even thinking about it.

Pronouncing everything is fine, might sound a bit strange but no more so than someone who never used contractions.

ahhh my favorite word to say! I felt so accomplished when I mastered it :P

す also.

There’s actually quite a few words that have the unspoken vowel actually built into their readings by replacing the second mora with a small つ. 学校 is the perfect example – just looking at the kanji, you’d expect the reading to be がくこう, but with the u-sound being suppressed, it comes out as がっこう, which is how it’s actually read.