The East Nashville area sounds perfect for you. It has excellent restaurants, shopping & nightlife; it's not a college community; most houses not only have yards but fences too, plus there is Shelby Dog Park nearby. In the hotter areas and/or homes with higher-end finishes, you'll have to pay more than $250 for a 3/2, but it shouldn't be too hard to find something that still meets your criteria.
A brief background: In Nashville's early days, around the turn of the last century, e.nash was the luxury area with beautiful mansions, brick sidewalks, ornate churches, and (a little further out - riverside, etc.) weekend/vacation homes. As commerce continued and crossing the river became a barrier, things moved west. In the 70's & 80's, during the withdrawal to the suburbs, e.nash became neglected, somewhat deserted, and overrun with drugs and crime. It was the bad part of town. Slowly but surely, original residents started taking a stand and other urban pioneers moved in to restore and reclaim this beautiful community. A tornado in '98 hit e.nash hard, but this fortuitously created a surge of renovations funded by insurance claims. Shortly after that, urban renewal was becoming a popular idea in major cities everywhere and varied people, but especially young professionals, came in to stake their claim on these beautiful properties, so close to downtown, and many people continue to work to restore and revitalize this eclectic community.
There are four historic districts. Closest to downtown & the large Titans stadium, there is Historic Edgefield. Some of the most beautiful houses are here (some worth well over a million dollars), but it's also partially adjacent to housing projects. Then, in the thriving 5 Points area, there is Historic East End. This is a smaller district with mixed-use zoning and some gorgeous homes. Next, there is the larger Historic Lockeland Springs, a larger district with lots of schools, churches, restaurants, coffee shops, bars, shops, and houses ranging from under 200k to $600k.
Those three neighborhoods are all adjacent. Then, about five to ten minutes further out from downtown, there is Historic Inglewood, a charming neighborhood growing in popularity, where you get more house for your money.