EDIT: Oops, I see Ashlucky has beaten me to the punch!
1) The masculine of prva is prvi, but I wasn't able to find the masculine of svetovna, what is it?
Svetoven (indefinite), stetovni (definite). [svetôven
Don't worry too much about the difference between definite and indefinite adjectives, the distinction only exists in the masculine nominative singular.
2) Is je bila like has been in English (but with bila declined for person/gender/number)? What's the infinitive?
is she was or has been. Bila is declined for person and number (feminine, singular). The infinitive is biti [bíti
]. The present tense conjugation follows a different pattern from different etymology: (singular) sem, si, je, (dual) sva, sta, sta, (plural) smo, ste, so.
Not to be confused with bíti - to beat, which conjugates as bȋjem, bȋješ.
3) What's the masculine? Globaln?
Quite a good guess since the vowel here is epenthetic and the root is indeed "globaln-".
The masculine singular nominative is globalen or globalni. [globálen
4) What's se?
It's a reflexive pronoun for all persons and numbers.
In this specific case začéti se
is the intransitive equivalent of the transitive začéti
Začela se je vojna - the war started.
Začeli so streljati - they started shooting.
5) I'm pretty sure it means "has lasted" but for some reason I can't find the infinitive, if I enter "trajati" in the dictionary, it gives no results
Which dictionary? The Slovar slovenskega knjižnega jezika has it.https://fran.si/iskanje?View=1&Query=trajati
Vanjo so bili vpleteni vsi večji imperiji
6) The highlighted part escapes my comprehension. Feel free to give as many explanations as you wish about this part, I know it's quite a lot of stuff.
vanjo - in her (i.e. in the war). The unemphatic accusative form would be "jo", njo is the form used with prepositions: vanjo, nanjo. The masculine or neuter equivalent would be vanj
vpleteni - entangled, intertwined, involved. From vplesti se.
Večji is the comparative form of velik.All the larger empires were entangled in it
.ter njihovi zavezniki
ter - and, as well as
njihovi - possessive form of "oni" (they).zaradi česar je sprva majhen evropski konflikt na Balkanu prerasel v vojno svetovnih razsežnosti
Zaradi means for the sake of, because of and activates the genitive.
Česar is the genitive form of kar (which). "Due to which".
Sprva - at first, initially.
Prerasti (prerástem) - to grow into. To grow is rasti, rastem.
vojna svetovnih razsežnosti - a war of global dimension, magnitude
razsežnost means dimension (SKSJ gives dimenzija as a synonym although I'm not sure to what extent this is used)