done some in nj ( 5 yrs ) where we had 12" - 24" frost
ok, here we go,,, 1 gal = 231 cubic inches,,, 30 cycles per min @ 2 gal ea = 8.02 cu ft,,, in 24 hrs, 193 cu ft,,, sounds like a 35-40 gal sump from apron store,,, & right now this pump drains out on the lawn ??? when snow melts, is frost still present & til when ?,,, unless you have gravity flow of discharged water, this isn't the good recycling you want to witness

original water source could be underground stream, surface water infiltration, or just hydrostatic.
a true french drain's a swale on the surface ( originated at fountainbleu, fr ),,, what you describe is a 27 cu ft cistern depending on soil percolation for efficiency & size,,, your box of rocks will hold more water if it ain't filled w/rocks,,, discharge this drain system down near the storm sewer inlet w/4" s & d pvc pipe,,, 1/4" drop in 10' won't ice up,,, to address your concerns: 1, yes - just dig hole & install 4" pvc; 2, how could that happen since its all below frost line ???; 3, done some

4, nfw

not for the amount of wtr you could have.
worst case scenario figuring you'd have no perc quality in the soil, this system pumps 1,440gal in 24hrs - that's 1537.33 cu ft,,, if running at your posted rate so you'd need a box measuring 10' x 20' x 7.685' ( no rocks, neither

) we ran into jobs where there was no slope to the storm drain - that meant digging & discharging below grade - you may have to have a permit - we didn't.
hope this isn't too confusing & helps ( yic-yac )