Welcome to the neighborhood! ;)
1. Groceries - As mrpoppinzs said, the new Hills Market is going to be a nice addition to the area, as it will be a 10-15 minute walk (or in our case, on the way between home and office). Great for speciality foods, prepared foods, and local Ohio products. Otherwise, we drive for most groceries as we're a household of four, and have 2 Krogers and 2 Giant Eagles within 2-3 miles. We mostly go to the Brewers Yard Kroger. And yeah, Arena Produce is a nice nearby option for fruit/veggies, though it's technically in Milo-Grogan and not really easily walkable from The KLD.
2. I'm probably the wrong person to ask about vegan stuff. Hal & Al's is a short drive or bike ride down Parsons and I think The Tavern might have some vegan fare. Of course, everything in Downtown and The Short North is pretty close as well. Tip Top is only a 20-25 minute walk when its nice out.
3. +1 to Los Potosinos (cheap) and Creole Kitchen (cheap) as well as jazz in the park in the summers (free), and walking 10 minutes to the Columbus Museum of Art (free on Sundays). There's quite a few public events (some free) at the King Arts Complex and CCAD, and once Hills opens there will be cooking demos, classes, wine tastings, etc (most events are cheap). The Arena Grand movie theater is a 10 minute bike ride and rarely overcrowded. There's also regular monthly art exhibitions at Cobenick Studios on Parsons (free). And plenty of regular events at the Columbus Performing Arts Center (cheap), a short 10-15 minute walk.
4. Best part of living in the area? While we could certainly use more things to do within the neighborhood itself, the ratio of affordability-to-proximity is a huge benefit for us. You can't get closer to Downtown for cheaper. And there is something to be said about being close to the bustle without being right in the center of it. Mayme Moore Park has a nice little playground that our kids use a lot. We love taking our kids to festivals and parks (Topiary, Franklin Park, Commons, Bicentennial, North Bank, Goodale, Schiller) all year long. Our neighbors are all pretty friendly and there's a growing group of people interested in improving the area back to its former glory. And we love working on fixing up a historic old home and investing into an urban center that is in need of some TLC. The more the better. ;)
Hope all of that helps!