I think my general answer to you first couple of questions is, if you're concerned about the sugar syrup going rancid, I'd stop feeding them. It seems to me if the sugar solution is sticking around that long, the bees have less interest in it than in other flora nectar sources available to them. It's time to stop feeding them. When you've got a hive that needs to be fed, they're not going to waste any time in sucking down the sugar syrup! I probably don't have a good answer about the bacteria thing; I tend to boil the water I use for sugar syrup - but that's mostly to get the sugar to dissolve faster (as opposed to trying to kill any bacteria in the tap water - however I'm sure it accomplishes that too). You may be able to research this question more on the internet (or some source like, "The Hive and the Honeybee").

There are many different types of feeders; each with their own 'pros & cons'. You can use a larger mason jar than a quart (if you can find the half gallon size in your area). Some of the bee supply businesses have a gallon size plastic pale that works pretty well - but again, if the bees aren't taking down the syrup pretty fast, mold and bacteria will grow in it - I've never noticed that caused a problem for the bees - but then again, I'm not looking down at the microscopic level. Some say the entrance feeder (i.e., mason jar) helps promote robbing between hives - I've never had a problem with that being the case. And with pan-type feeders, you have to provide the bees a means to get to the syrup without getting in it and drowning.