I've done my research for a while, read some chapters on books about it, and I can't for the life of me resolve these problems. I have read the rules of this kind of algebra.
My teacher teaches this method kinda too barebones, I see everyone include much more intense procedures (things like the sigma and R).
My teacher says there's no "real method" to solve these problems.... which is driving me off a cliff. These problems are translated by me so the grammar will be a bit off, but please bear with me.
For example:
Find a RE which generates strings that contain "BB" over the alphabet ABC
He arrives at the conclusion:
$(a\cup b \cup c)^* bb (a\cup b\cup c)^*$
why? how does one do it?
Another one:
Find a RE that generates strings with the exact same number of B's and A's over ABC
$(ab\cup ba) \cup (ba\cup ab))^*(aa\cup bb)^*$
Why? if anyone has a source which deals with problems like this then please share it as well.
I agree with your teacher. There is no recipe for producing regular expressions that generate a particular language. It's often difficult even to decide whether such a regular expression exists. You have to study the examples and work the exercises to gain proficiency.
The first problem seems straightforward to me though. We know the string has to contain $BB$ somewhere. We don't care what, if anything, comes before or after it. That leads directly to $$(A\cup B\cup C)^*BB(A\cup B\cup C)^*$$ You just have to realize that $(A\cup B\cup C)^*$ matches any string at all over $\{A,B,C\}$.
As to the second problem, I don't understand it. As I mentioned in a comment, this is not a regular language; there is no regular expression that generates it.