It should still be OK then. Car batteries last roughly five years
under normal use.

Oh yes! When the temperature is down, all chemical processes slow
down. If you look at battery specifications, you'll see a spec called
"cold cranking power" which is the ability of the battery put out
decent current at low temperatures.

Your problem however sounds more like cables and connections. Here's
an excerpt from the FAQ

"Then again there's always the possibility you have bad battery
connections or dirty solenoid contacts. Here's how to check. First,
loosten the battery terminals a little and move them a bit, then
retighten. Check where the negative cable is bolted to the body or
engine. If the engine isn't turning and you just hear a clicking sound
from under the hood, turn the headlights on and have someone observe
them while you try to start the car. If they go out, the battery is
being loaded down and can't hack the load. If they stay lit, the
starter solenoid contacts or the starter's comutator is dirty and the
starter isn't drawing its full current. The latter fault is often
intermittent."

More at http://www.tegger.com/hondafaq/faq.html#battery