"I want just to avoid problems" means the same thing and should be accepted.
Yes, there seems to be nothing wrong with that and I wish someone had responded to you.
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
"The problems" can refer to aforementioned ones not restated in this sentence.
How do you distinguish between "I just want to avoid problems" and "I want to just avoid problems"?
I just want to avoid problems = je veux seulement/juste éviter les problèmes (= my only will/wish)
I want to just avoid problems = je veux éviter seulement/juste les problèmes (= the only things I want to avoid are 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.
What's happened to the preposition 'à' for the infinitive? Does 'juste' act as the preposition?
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.