Above comments are correct about the fumes, I've read there is no problem trickle charging as long as the battery is free to vent. However, if the battery does fully discharge during usage, I may hook up a larger charger to bring it back up in a shorter amount of time, and will aim a small fan at it to disperse any gasses.
115v inverter will be fine, I would stay away from the 110v ones though (sounds like they are being cheap). Lower voltage produced means the current draw will be higher to get the same power. This means if a motor is rated at 1A at 120v, then at 110v it will draw 1.1 amps.
I was thinking the same thing as GreatLakes about using the larger inverter, I would rather the thing coasts along driving my pump than pushing it's rated limits. Plus, you could always add a second battery to extend run time/load capabilities without much trouble.
A far as the outlet ground (good question, gave this a lot of thought!) The house ground is tied to the output ground directly all the time (doesn't go through the relay). This means the inverter ground is not tied to anything. Even when the power fails, your house ground still works
