Does the divorce that I swore, intending that I would not speak to any woman, take effect if I then spoke to several women on social media? And if it takes effect, is it necessary to inform the wife of the divorce, or is it permissible to take her back without her knowledge? And what is the nature of the re-establishment of the marriage?

The wife's refusal to return home, conditional on an oath, is not considered a valid form of duress. The breaking of an oath depends on your intention at the time of swearing it. If you intended by your oath never to speak to a non-mahram woman and you broke it, then a divorce occurs. If this divorce does not complete the three divorces, then you have the right to take your wife back during her 'iddah (waiting period). The validity of the return does not require the wife's consent or knowledge. It is completed by your saying, "I have taken my wife back," or something similar. You must fear Allah and stay away from illicit relationships.