The answer to the first question is yes, て can be used to combine the conditions.
As for the second, it does sound like a sequence, but it is by no means strict and "if AしてBしたら" is understood simply as "if A and B, then...".
Two exapmles (with unnatural literal translation):
お風呂に入って歯を磨いたら寝なさい If you take a bath and brush your teeth, then ...
I was wondering why there is a を there. I thought that if the first verb is intransitive, and if the second one is transitive you could put a を?
Right. Here 土日 is the object of transitive 過ごす. Please refer to:
How can verb て become an adverb?
Can the を in ～を通して be replaced with は?
From @naruto's answers in the pages above:
the first verb before て ...
The core verb here is 過ごす【すごす】, which means "to pass or spend [a period of time]; to pass over something". This is the transitive form of 過ぎる【すぎる】, which means "to pass by; to surpass; to be excessive".
As a transitive verb, 過ごす【すごす】 can take a direct object. In your sample sentence, this is 土日【どにち】 ("Saturday and Sunday"): ...
The た-form would be correct if the sentence ended there.
The compound verb 勉強してくる is used in its て-form to connect these two sentences into one, in one of the most basic functions of the て-form.
The sentence sounds a bit unnatural to me, though. I would probably use an adversative conjunction.