I suspected my wife of communicating with a foreign man, so I made her divorce conditional upon this three times. Then I divorced her again and took her back. I am asking about the ruling regarding the first conditional divorce: does it count as two divorces or one, and is she forbidden to me despite not knowing if the condition was met, and where is it preferable for her to reside?

All praise is due to Allah, and peace and blessings be upon the Messenger of Allah, and upon his family and companions. To proceed:

It is obligatory to think well of one's wife, and it is not permissible to think ill of her without evidence. If divorce by triple talaq is conditioned on her communicating with a strange man, then the divorce occurs upon the fulfillment of the condition, according to the majority of jurists. Some jurists, however, hold that if the intention was merely to threaten, then an expiation for breaking an oath is due. The pronouncement "You are divorced, divorced, divorced" counts as a single divorce unless one intended three. The phrase "without return" is considered null and void by the majority. If her communication with a stranger is not proven, then the default is that she remains under his marital bond. As for the waiting period ('iddah), a woman who is revocably divorced must observe her 'iddah in the marital home. However, for a woman who is irrevocably divorced, the majority of jurists allow her to observe her 'iddah wherever she wishes, but if she observes it in her husband's house, she must be in a separate section.