These translations can be tricky, mostly because place perspective can be more flexible in English than in Portuguese (for instance, it's more common in English to mentally put yourself on the place/situation you're talking about). For example, from the WordReference (my emphasis):
I will bring the car over to your house if you drive me home afterwards.
Eu levo o carro até a sua casa se você me levar até em casa depois.
Here you see that the speaker assumes their interlocutor's perspective (assuming they aren't together at the latter's house) in English, while in Portuguese it's usually more natural to use "take", sticking to your own perspective.
This flexibility is nicely described in the American Heritage Dictionary's (via TheFreeDictionary) Usage Note:
The difference between bring and take is one of perspective. Bring indicates motion toward the place from which the action is regarded—typically toward the speaker—while take indicates motion away from the place from which the action is regarded—typically away from the speaker. Thus from a customer's perspective, the customer takes checks to the bank and brings home cash, while from the banker's perspective the customer brings checks to the bank in order to take away cash. When the point of reference is not the place of speaking itself, either verb is possible, but the correct choice still depends on the desired perspective. For example, The labor leaders brought their requests to the mayor's office suggests a point of view centered around the mayor's office, while The labor leaders took their requests to the mayor's office suggests a point of view centered around the labor leaders. Be aware that the choice of bring or take determines the point of view emphasized. For example, a parent sitting at home may say of a child, She always takes a pile of books home with her from school, describing the situation from the child's viewpoint leaving school. If the viewpoint shifts to the speaker, bring becomes appropriate, as in Look, I see her coming right now, and she's bringing a whole armful of books!
trazer = “bring” and that levar = “take”
is a good rule of thumb, but it'll work best if you remember to keep the speaker's own current perspective.
And finally answering your main question
Can I use levar in any situation of bringing something from one place to another?
Mostly yes, as the dictionary attests: while the first meaning of "trazer" is (my emphasis)
Transportar para cá
[to transport [it] here]
the first meaning of "levar" is simply "to transport":
Fazer passar (de um lugar) para (outro); transportar
[to make it go (from a place) to (another); to transport]
that is, in principle "levar" is more general, neutral (though most often it's not used when this other place is "here").