I recommend the 11th arrondissement. Some of that area is getting a bit hip. I spent a day walking around there, searching for some of the hot coffee shops (I did find one but the name is escaping me right now). I did go to a high spot in Paris, not sure where, but I had read about it in an online article. I walked up and up and found a lovely small park with a good view over Paris. It was in or near the 11th. perhaps search for high elevation point in Paris and you too may find it. Great tiny streets and walkways. It was outlined in my map book too.
David Lebovitz (chef) recommends the Barbes market which is multicultural and could be overwhelming with the jostling. A bit tamer is the Aligre market. He has a website and did just post about this on FB a day or 2 ago. I find the food markets farther from the centre are pretty cool.
I visited the Commerce market in the 15th, so a bit tonier, but a lovely market with none or very few tourists.
I stayed in the lower 5th a couple of times, near rue Mouffetard and visited place d'Italie in the 13th, which not gritty is more regular people world. when staying on rue Claude Bernard, we breakfasted with the locals at plain old cafe on the same street, with the old dudes hanging at the bar sipping coffee (and maybe wine), some students and older couples meeting for a coffee. It was a nice environment. We definitely were the only tourists at breakfast.
I would say just roam farther and farther from the centre and see what you can see.
Enjoy Paris, I am counting the days until I will be there! June