I'll give some quick, short answers first ...
Seller Disclosure (as DH points to) is required in some states and is mandatory in others. Depending on the laws of your state, you could be at a loss. EVEN if your state required it, and the owner 'lied', you'd have to prove that they knew the basement leaked. The presence of a perimeter drain system and a sump pump don't mean that the basement had leaked. It may be shown that they were installed 'just in case' the basement leaked. In short, trying to prove the previous owners knew 'something' may cost more $$$ than any recovery will pay.
RE: Biblical Rains - It's quite possible that the Biblical Rain you're refering to was the rain that 'broke the camel's back'. There MAY have been Biblical Rains PRIOR to the sale of the house but proving that they 'infiltrated' the basement may be hard to prove now.
The styrofoam lining may be part of a perimeter drainage system. Look for 'channeling' in the foam or channeling on the floor of the basement. All that's, essentially, done is a way is made to 'channel' excess water to the pit. If the foam doesn't have anything to do w/a perimeter drainage system, I can't tell you what it is. I DOUBT the foam has anything with removing leaks through the walls ...
RE: Trapped Water under the 'Drylok' - If the water doesn't run 'down' the wall and out the bottom, it will eventually 'blister' through the Sealer. So far, you're lucky that the water isn't pushing through the Sealer. For now, leave well enough alone on this ...
Did you have a Home Inspection? Please answer this ? for me. If you did, what did the Inspector say about what you're asking 'us'? (You said you just bought the home, so you should be allowed to call him back if you don't have a 'good answer' to your question(s).)
Lastly, 95% of water-in-the-basement problems are associated with leaking gutters, missing gutters, improperly INSTALLED gutters, missing or improperly installed downspouts, water-dumping at the foundation, and wrongly-sloped landscaping around the house. The next time you get a heavy rain (and you don't have to wait for a Biblical Rain), don your raincoat, boots, and an umbrella (watching out for lightening), and look UP and DOWN. Look at both. See if you have leaks, excess runoff on the roofline, watch where the water is running to or collecting at, and so on. The KEY to fixing these problems is relatively inexpensive. Sealing up walls and installing perimeter drain systems and sump pumps are usually 'bandaids' on a forever bleeding wound. Over time, water that is 'permitted' to leak into the basement may be undermining the foundation, or causing cracks to enlarge, or who knows what else. FIXING the problem from the outside should be, IMO, the FIRST line of defense.
Take this all to heart and you should do much better in the future. Please tell 'us' what the Inspector said about your ?'s. My best to ya and hope this helps.
Jay J -Moderator
PS: God Bless America!