Just judging from the greenery vs the pot, I'd say, sooner or later, it needs re-potting.

I would pot it up, but only in a pot slightly larger, palms prefer being somewhat snug in their pots. But this also resents drying out, so keep up with moisture levels.

Any thoughts on that sphere-like object, as pictured?

Its the old seed coat (testa) that one of the seedlings sprouted from.

Safe for me to leave the testa where it is? Also, what about the white splotches on the leaves?

The splotches are just marks from overhead watering at the nursery. Not an issue. Leave the testa alone.

I have few Chamaedorea species..., all terrific indoor palms. I have C. elegans (Parlor palm), Bamboo (C. seifrizii), C metalica, and Cat palm.

The size of a plant above the soil is not a reliable indicator of whether it needs repotting. A better guide is the ratio of the roots to the soil. The easiest way to determine that is by how often the plant needs to be watered thoroughly. Only if it needs more than twice-weekly thorough watering does it need a pot one size larger.

This plant is very prone to spider mite infestation. Check very carefully for fine webs and/or dust-like particles on the leaves.