Sorry ta hear about your situation, I`ve heard this so often it drives me crazy.
First,from what you`ve posted, the Real Reason(s) water in entering is because 1)there are crack(s) in the Bsmt wall which is How `n Where water enters the 'cells' of the block wall 2) the window you described will probably need to be replaced 3)there may also be openings around where the gas line-pipe enters the house,water gets into the block where many lines enter a wall,just like through crack in a wall.
Something you may/may not know about block walls `n that is on all block walls not ALL cracks are visible from the inside of the basement. Meaning...just because you see some cracks does not mean there arent others of ANY size on the outside of the walls, where water-moisture enters. In otherwords, a home can have other hairline cracks and even 1-2" wide cracks on the outside which do not show from the inside in the basement.
And you mention the soil in your area is very sandy which tells me thats the ONLY reason you dont leak on less heavier rains. Because sand is the 2nd best backfill for drainage along basement walls..peastone/gravel is best.
When there are cracks in a basement wall the only way to stop water/moisture from entering through these carcks is from the outside,Not inside. The wall or part of the wall say,where a vertical crack is needs to be hand dig to footing/Dr. tile, haul all soil away, wire brush `n scrape wall of any excess dirt,seal all cracks & other openings(around a gas line etc)with hydraulic cement, then apply a Thick tar over entire area of wall `n then apply 6 mil+ visqueen over the tar and backfill with 100% peastone/gravel to within 3-6 inches of grade,fill remainder w/top soil....will continue.............