Had a great time here, El Born is an amazing part of the city and this apartment is right in an awesome little square with restaurants, musicians, children playing, etc. For groceries and supplies I recommend heading over to De tot al Born, which is open 7 days a week and lives up to its name and has high quality stuff. Xiloteca is a great little specialty coffee shop right around the corner and just a little further there is an excellent bakery Forn la Llana.
The air conditioner was replaced early on in my stay and works great (though it was mostly unnecessary in June). In general I thought it was fairly clean (minus some of the kitchen stuff needed a good clean, but I would do that anyways). There were flies and what not, but I couldn't find any evidence of problematic bugs like cockroaches/bedbugs/etc.
The building is old and the neighborhood is noisy, so if you think that will be a problem for you probably want to look further away from the city center.