Omission if information is something to be concerned about. Of course, unless you have prove that something more is going on, for the time being try to just observe and keep your guard up. Ask him how would he feel if you did that and if he can see your point of view.
The insecurity is something internal. He can not make it better or worse. It is something that you have to control (you already realize that it is problematic for you. Monitor your negative thoughts or anxiety. If you seem clingy or too insecure, he may be turned off.)
Men can have female friends. You may want to figure out what do the two have in common, what is his motivation to seek her company, etc. Having lunch is not an indicator of romantic interest (it also depends on what kind of place they ate at. Some fast place joint or a upscale restaurant)