I think it might be helpful to distinguish between the fact that he contacts you in a friendly way and the fact that he flirts with you.

He might flirt with you because he thinks it is innocent fun and imagines that you do as well.

On the other hand, he might flirt to keep open the possibility of connecting more closely with you. This could mean that he is genuinely unsure about you versus the other woman, or it could mean that he is keeping you handy in case other options fall through.

Many guys will keep a woman handy if the woman doesn’t take steps to stop it.

He might just be lonely, but flirting isn’t needed if one is simply trying to address loneliness.

It is nice of you to help if he is in fact lonely, but I was glad to hear that you are taking care of your own feelings as well. As you know, a person with unreciprocated romantic feelings can be hurt while acting as a platonic friend or de facto therapist.

You mentioned holding the door open for him because you aren’t interested in dating right now. Must you put up with confusing flirting in order to do keep the door open?

I wonder too how much easier it might be to move ahead in your dating and regular life if the tantalizing prospect of him were not regularly floating before your eyes?