Bismillah. So it sounds like there are two dimensions to your question. One is legal and the other is a matter of preference. The legal question is what is the level of interaction that men and women can have with each other if they are not married and not mahrem to each other (and both are past puberty). I think you will find a range of opinions on this issue among the scholars.
Sadly there are not many real scholars here on stack exchange so I will try to answer this question as best I can (but please bear in mind that I am no scholar and so this is just my opinion).
Although there are text about lowering the gaze, there is no explicit text which I know of that says non mahrem men and women cannot speak to each other. I also know of at least one hadith where male and female companions were speaking to each other, I can't recall all the details off the top of my head but it was a discussion some companions were having about who had served the Prophet (saw) more, be it those who were with him or those who were in Abyssinia. There are probably several other examples but I can't think of any at the moment.
So this seems to indicate that it is not haram for non mahrem men and women simply to speak with each other. Now the next question is can they have prolonged conversations, like friends, over a cup of coffee. Many would say this is free-mixing and is not allowed. I tried to follow Shakjh Faraz's reasoning in this post and he doesn't seem to bring much in terms of explicit text but it seems to be grounded in the principle of blocking the means (saddal-dhara’i).
But I think some would bring the argument that it is possible for a man and woman to be just friends and not fall into zina. It just depends on the people.
So my humble opinion is that a situation like you described might be permissible, especially if it is always in public (because there are explicit text about 'being alone with each other'), you do not fear that it might lead to zina, and your intention is this person should get a better perception of Islam and Muslims (dawah).
The second dimension of your question is preference. Some people are more comfortable interacting with the opposite gender than others. I know my wife was brought up in a very conservative culture so she just doesn't feel comfortable making 'small talk' with non mahrem men. I respect this and do not force her. But if your preference is to have such conversations then please go ahead, but realize that many Muslims may find this awkward based on their culture and upbringings (which tend to have a very restrictive and conservative view of Islam).
As a final note, I would just add a word of caution that if any in any of your interactions, you feel an attraction building then you should check yourself. And in that case, you should evaluate if that is real love (in which case you should consider marriage) or just lust (in which case you should fear Allah and limit your interactions with that person).
Allah knows best.