Are you consistently questioning, lacking trust, and showing signs of paranoia? Is there any past behavior from him that provided concrete evidence for your distrust? Could it be that you have deep-seated trust issues stemming from previous relationships?
It's possible that she signs all cards in a similar manner as a friendly gesture and a way to show appreciation for those attending her class. Perhaps even those of family and friends as well.
Perhaps he isn't necessarily going to the gym early; maybe he goes somewhere to enjoy a meal or other activity. It's also possible that he goes to the gym early to socialize or because he values getting a good spot in class. Alternatively, he might simply enjoy his alone time elsewhere.
Maybe he didn't disclose that the gift was from her because he's wary of your tendency to overreact and jump to conclusions.
There are numerous potential explanations.
It's possible that neither of them finds the other attractive.
Conversations between men and women don't always signify attraction; not everyone is universally appealing.
The card might be displayed because he genuinely likes it and appreciates the gesture, especially if you haven't expressed any negativity towards it.
Without solid evidence, it's unfair to assume there's something romantic happening between them.
Consider giving him the benefit of the doubt until substantial evidence suggests otherwise. If that happens, address it accordingly.
If you consistently exhibit paranoia and mistrust, it might be best to reconsider the relationship. Continuously doubting him isn't healthy or fair, particularly if he's innocent.