I start bees in stacking 5 frame boxes all the time, including medium depth nucs. When the colony is established, the 2x5 frames graduate to a 10 frame box. A week or so after the stack is returned to one (bigger) box they are ready for supering. The colonies really take off under this management.
The query you made, "if a recently hived package should be resplit on June 1" is separate question. This is only going to answered with local experience. A "walk-away" style split is going to be about 45 days (or July 15) from its very first cohort of bees from the new queen. A requeened split shortens the period to late-June.
Many areas (including mine) experience a mid-summer dearth, and wax and brood production drops to marginal levels. If you are splitting into the dearth, the hives will be set back strongly, and you end up with little colonies that never, ever take off.
If you are trying to manage these little nuc sized colonies in the same apiary (or foraging territory) as production stacks, you will end up with massive robbing issues during the summer and fall dearth periods.
A new beekeeper should not try everything at once. One cannot learn if one is fiddling with all the variables. One should learn what the local period of surplus and dearth consists of. You do this by watching the behavior of undisturbed colonies carefully. Armed with that knowledge, one can plan one's increase for the subsequent season to build into the period of abundance.
Splitting into peak flow makes everything easy --- queens seemingly appear out of thin air. Trying to get wax drawn and brood laid during dearth requires epic efforts, some of which precipitate robbing.