The pot is about 6 and3/4 inches.I'm not the best
plant person in the world so am impressed that they got
this far.

If the pups are nearly as large as the original plant, then they should bloom this summer. I put all my broms outside for the summer under 40% shadecloth starting in early-mid May(or after danger of frost). I think that the fluctuations in daylength and temperature naturally trigger flowering for many bromeliads. My Aechmea fasciatas bloom quite reliably during July and August. How crowded to allow them is partly a matter of taste. A 4 inch pot for 3 pups sounds way too small for me. You could repot the clump as is, in a 7-8 inch pot. I sometimes allow 3-4 in a pot, but I usually prefer individual specimens of wide leaved broms. An east window should be fine for now. If you decide to try moving it outside for summer, acclimate it slowly to sun or it will burn. Try a northern exposure for starts.