Answer: I suspect you are seeing one stage in the life cycle of the fungus gnat which looks very much like a fruit fly. These breed in moist soil and can be eradicated by allowing the top layer of soil to dry out between waterings. This can be difficult when you are trying to keep a baby plant evenly moist but it may also be an indicator that you are overwatering. Another approach is to place a free-draining thin layer of gravel or sand on top of the soil. This should stop subsequent generations.
Spider plants can develop brown tips for several reasons A very common one is underwatering which can in turn be caused by a need for repotting. Be sure that the soil is truly wet when you water and that the water is not just running out between the soil and the edge of the pot. If the soil has become very dry place the pot in a sink or bucket of water and allow it to wick up water until the surface is moist, then allow to drain thoroughly.
Another possible problem is salt accumulation in the soil. The soil should be leached occasionally to wash out any accumulated fertilizer salts. Do this by watering to the point of free runoff several times in succession.
These plants are not terribly fussy about their fertilizer requirements so any balanced water soluble fertilizer for foliage houseplants should do. Be careful to apply according to the label instructions; more is not better when it comes to fertilizer.
Finally, dry air and drafts can also cause brown tips. Trim them off with a scissors and the damage will be less noticeable.
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