I want to know:
1. Did I commit a sin by talking to her under a different name?
2. As I made her think that the person she was talking to was not her husband, did I encourage her thereby to commit the sin of talking about private matters?
3. As there was no other person in fact, will I be questioned about my pretending to be someone else?
I hope that you can answer these questions, because I feel that I have sinned and I fear Allah in my heart.
One should think positively of a Muslim and assume that he is good and innocent so long as there is no reason for doubt and suspicion, because Allah says (interpretation of the meaning):
“O you who believe! Avoid much suspicion; indeed some suspicions are sins. And spy not…”
This Qur’aanic etiquette is one of the means of attaining happiness and peace of mind, because thinking badly of people leads one to seek out their faults and may lead to spying or doing something like what you did, which was a mistake because it may make the wife dare to speak to men about private matters, as you mentioned.
With regard to talking to her under a different name, it is not a sin in and of itself. But we should speak about its motive and what may result from it. If the motive was thinking negatively of her with no evidence or grounds for doing so, then this is contrary to what Allah has enjoined of avoiding suspicion.
Trying to make the wife commit some sins was also a mistake, even if the intention was to test her. If we assume that she responded and did fall into sin, that would not have proven that she had gone astray or committed that sin before that, because a person may be weak at any moment in the face of temptation.
Thus it is clear that this action was of no benefit; rather it may be harmful and may encourage the wife to commit sin, and that would increase your doubts without enabling you to find out the facts of the matter.
It would be better to take care of your wife by getting in touch with her and visiting her, and praying for good things for her, and trying to increase her faith and piety, and letting her live near your family or some righteous neighbours. These are means of attaining righteousness and keeping away from evil.
Even more important than all of that is not being away from her for a long time for the purpose of amassing money; rather you should keep your journeys as short as possible, even if that results in more expenses or reduces your earnings. Being keen to take care of your wife and keep her chaste and give her her rights is more important and more obligatory.
And Allah knows best.