Reminder: "les problèmes" can mean "problems in general" or "the specific problems".
In this sentence, either "problems" or "the problems" is acceptable, without context.
"des problèmes" is the plural of "un problème". I guess that if the French sentence has "des problèmes", as in "je veux juste éviter d'avoir des problèmes", the English translations would include "I just want to avoid having some problems" (as more than one, or an unknown number of them).
"avoid the problems" isn't really correct here, because in this sentence both "les problèmes" and "des problèmes" translate to "problems", but with nuances in meaning:
-éviter les problèmes means that you want to avoid problems altogether
-éviter des problèmes means that you want to avoid some problems, a indefinite but limited amount of problems
That's not true. If the only things I want to avoid are problems, it would be "I want to avoid just problems." Really "I want to just avoid problems" means just about the same thing but is considered incorrect by grammar snobs because it splits the infinitive "to avoid." I would accept it, though.
There is never a preposition between the verb "vouloir" and another verb in infinitive.
"Juste" is an adverb in this sentence.
This is the list of verbs which can introduce an infinitive without a preposition: Aimer/aimer mieux, aller, compter, croire, courir, daigner, descendre, désirer, détester, devoir, entendre, entrer, espérer, faire, falloir, (s')imaginer, laisser, monter, oser, paraître, partir, penser, pouvoir, préférer, prétendre, rentrer, rester, retourner, revenir, savoir, sembler, sentir, sortir, souhaiter, valoir mieux, venir, voir and vouloir.