This actually depends on what UPS type you have, as there is several different types. Some allow this functionality, others don't.
It is possible that in your case if you take out the battery that it will work as you want. Ive seen in the past that sometimes when a battery fail that it has a failure inside causing the issue you refer to, then when we removed the battery the UPS worked again, of course it would have shut down if the power went down, but at least it allow you to make certain it remove any spikes in the grid until you get the battery replacement.
In regards to battery life, it mainly depends on the operating temperature. The higher the temperature, the shorter lifespan they will have. You will especially see this if you compare how long runtime a new battery have, compared to the runtime it has after one year.
As a side note, APC has a bonus program if you return a old UPS. I think you get either 20% or 25% off on the price of a new UPS. We have always used this instead of buying new batteries, and at the same time we get the extended warranty, giving give us a operation time on five year per UPS. If you browse some around on their website, I am sure you can locate this program.