This is not specific to the original post but my experience with storage and batteries...
Some trickle chargers or 'pulse' chargers may indicate 5A, 15A but it's not an accurate current load charging them. They use pulses that simulate the amperage shown. Basically, they indicate the 'charge rate', not the real current applied. Many chargers including the famous green
Battery Tenders can fail from cold weather use yet look like they're working. Chargers with lamps and no display are a mystery at best what's actually going on. New chargers are amazing, some even operate at freezing temps. Many chargers 'hunt' trying to determine if a DOA batt. is 6V, 12V, or what? They can't auto-range that dead thing. A cheesy charger will not pull them back unlike old-school chargers that arced and sparked when you hooked them up. They sent true 14 Volts at low current (1.5A) not pulses. Proper new chargers have manual 6/12V setting to 'force' the charge on a dead batt. If you don't need 6V, don't buy one that does both.
Old batteries or even new batts. stored in freezing temps are suspicious. Make sure and check the batt age. Four years or more is their end of life. A fresh batt. is better insurance than a tired unit pulling on the
alternator or charger constantly. AGM glass mat bats are correct for these cars. The $40 cheaper lead/acid options are not worthy for trunk stowage. Ask a pro.
These cars are very sensitive to have proper voltage present for the E-modules to communicate on start-up. You should have 13+ V at the batt before turning the key. Some higher end chargers can 'jump' the car where trickle chargers should never be connected attempting to crank the car. Charge the thing with a known good charger and if it's flakey jumping around pull the batt and have an auto store load test it. Cheap chargers are junk and not trustworthy after a few years. The light and horn show is due to low power and/or module reset, it's basically confused. Scared the hooey out of me setting off the alarm swapping yes, a battery that went to junk a few weeks ago. Note: the keys or fob are handy to disable the alarm when you set it off swapping batts. I had to dash back to the house to get them to shut the thing up! Window reset and other tweaks are needed after a batt. discharge or pull. Once you know the actions and reactions disconnecting the batt, you can plan for it.
I have a $60 pulse charger and a new $170 AGM batt. The charger is needed every 2-3 weeks to pull it back up if left sitting in-car. Meaning, a couple months sitting will pull it down to zero. In freezing winter, the batt. is stored and charged indoors. Charging is only needed a couple times over winter storage. Doing nothing and letting the batt die thinking you'll just charge her up in the spring might be a practice that worked but at what these batts cost, a bit of effort can make them actually last five years. Sitting dead or frozen is not good. You can connect charger gator clips under the plastic lip above the Neg. terminal. Just pull the spare cover and sneak the clip under the lip and charge 'em. Easy - Peasy!