The technical term is bulbil, and it is how some lilies multiply. For some types of lilies, bulbils are produced every year, while others grow them only under certain conditions and usually do not produce bulbils. Unlike seedlings, bulbils are exact replicas of the mother plant, and will produce the same flowers. You can remove them when they are ready to fall off by themselves, and plant them a quarter to half inch deep. Leaves may or may not emerge this season, but will definitely be produced next spring.

You could repot the plant, which is what I would do. The reason is that I would want to know what kind of potting mix the plant is growing in, rather than let it remain in the unknown mix which came from the store. Sometimes I also feel the mix they provide is some cheap mix, just enough to get the plant on the shelves and into someone's ownership, where the eventual death of the plant could be from many reasons, but the owner will sometimes think it is because they didn't know how to take care of it. The conditions you described could have several causes. In general, make sure it has enough lighting and don't overwater or underwater.

Unfortunately, you can't grow an Easter lily as a a houseplant successfully. Plants that normally grow outdoors in temperate climates - like the vast majority of lilies - do not make good houseplants, as indoor growing conditions fail to provide sufficient light, humidity and most importantly, the low temperatures necessary to accommodate a chill period or natural dormancy. You could possibly allow the bulb to go dormant (all the foliage dying back) then chill for the appropriate amount of time in the refrigerator, but you need to avoid fruits and some veggies stored in the same frig as they give off a gas that will impair blooming. And with no guarantee that the lily experienced adequate photosynthesis to generate enough energy to rebloom. The insects in the soil are most likely fungus gnats or their larvae, a common pest of indoor plants due to excessive watering and a poor quality potting soil. They feed on plant roots and decomposing organic matter in the potting soil. If I were you, I'd give the lily to someone who had a space to plant it outdoors. And treat yourself to fresh, healthy and budded Easter lily when they are marketed in the spring to enjoy while in bloom and then discard.

Where are you located that Easter lilies bloom in March?? That is NOT typical!! These are a summer blooming lily (Lilium longiflorum) and their standard bloom time is June and July and maybe as late as August. Blooming in March is not the normal time anywhere! They only bloom at Easter if forced to by the specialty growers. I've grown many Easter lilies over the years, both from dormant purchased bulbs and from recycled living plants received as gifts. One planted in the garden, they will bloom at their normal time of midsummer, even in my very mild climate.