Unfortunately, a woman's intuition is usually correct. If you have to ask yourself this and things don't seem right, then they are not right. Any guy that would play games with you like this and make you worry and wonder what is going on is not worth your time. Cut your losses and move on.
No Why would it be pushy, just ask him. If something is bothering him he may want to get it off his chest anyway but feels weird confiding in you for it has only been a month. Heads up thou.. If you don't want to be a possible 'cry shoulder' shut the 'F' up and don't open those doors, let someone else do it for him. You will still reap the benefits after he lets it out.
no weirdness here at all
That's not pushy at all. If anything it shows that you care.
His true self is starting to show. Move on.