But right now a trickle charger isn’t really what you want for best results. A decent quality “smart” charger that works with AGM batteries would be best. Either way, charge the thing up and see what happens. But a trickle charger won’t set anything on fire.
You don’t need to remove it from the car, although I always do – it’s just a preference. I don’t want my trunk open or my charger cables closed up in the trunk lid. But I also don’t use a trickle charger.
Do you have a voltmeter? Get the thing up to full charge. Drive the car around for a week or so, and monitor the 12V battery voltage each time you let it sit over night or so. A healthy, original AGM battery should have a “resting charge” (after sitting) very near 13V (figure 12.7-13V). If you consistently find it sitting below…[very fuzzy number]…12.5 or so, then suspect that the battery is nearing its end. If you find it sitting nearer to 12.0V – first worry about a parasitic current draw. And then worry about the battery.