In actual practice on my cheapie PSUs, the side effect of leaving the battery connected to the output has ranged from 'an led stays lit forever' to 'the cooling fan on the psu will intermittently run for no appreciable reason'. But I have yet to break one from 'leaving it hooked up'.
As far as whether or not to leave it hooked on the AC side, i HAVE blown up one of the 'inrush current limiter' thermistors on the input side. Theoretically that shouldn't happen to begin with, but if we acknowledge that it is possible for it to happen, leaving the unit hooked up to the AC source should prevent it. But you may still run into 'fan runs for no reason even though psu sits idle' and you will have some small idle draw.
As far as protecting the output, you can put a switch on the output circuit, and can even control that switch based on voltage levels etc. I've posted elsewhere about using a cheap PSU as the input to a cheap pwm SCC that has a current limiting feature, at which point you have something close to a 'real' battery charger with a ton of adjustability for ~$40 on the low end. The SCC will simply disconnect the PSU from the battery (or the negative pole of the circuit, at least) when voltage rises to the setpoint you create. You could add another device to make it disconnect the other side of the circuit too, but there would be some minor caveats. Or just use the load port of a cheap SCC to control a switching device directly between the psu and battery without making it flow through the SCC. This is just using a <$20 pwm scc as an adjustable 'voltage window switch' between two devices. I have bought adjustable voltage window switches that ARENT scc's for as little as $5 on Amazon, too.