Welcome, Jenny! I'm not seeing a video in the thread, only jpg pictures.
Thinking along James's lines, I did a quick websearch with my limited vocabulary on the subject trying to find a more expert opinion on some of the challenges you might face with possible solutions. That led me to this UK article about historic wet cellars which may be of use or not:
Damp below Ground Level in Historic Buildings. Did you pull permit records on the place yet to get a better understanding of what possible water intrusion mitigation work was done aside from the stepped retaining walls on the sloped sides and the PVC drain? (Either the town or possibly county offices will have these.)
Speaking from personal experience, at one point I was a property manager for an 1860
servants' quarters that had been split into three apartments. It too was built into a slope, but on the long west side of the house. There was a stepped up portion to the yard side. It was up from the door threshold level walkway, but it was slight and not really a retention wall. The south short side stairwell has one retaining there. The east walkout from the cellar kitchen (with still intact cooking fireplace and beehive oven not shown in the listing, but the main floor cooking fireplace is) is on the lower end of the slope facing towards the main house. The quarters' property had been parceled off from the portion with the main and carriage houses. You can see bits of the carriage house in some of the pics, but not the main which is now a funeral home with the paved parking lot right bang up level near the quarter's lower exterior doors. If there was a spring on the grounds, I was never informed by the home owner. I do know there was a cistern somewhere on the property but was never shown where. The cellar did have damp issues, but I suspect because the upper portion of the slope was level yard with one step down for the walkway plus possibly the cistern up there as well helped mitigate moisture to a degree where there was never outright condensation dripping after heavy rains. The oil burners and electrical boxes were in the cellar in a small room behind the oven wall (likely the wood room back when). Only once while I was there did we have minor water intrusion in the cellar through the east parking lot side exterior doors from flooding rains, but the water really had no where else to go since there is no slope down from the doors to the paved parking area as you can see from the photos, and no french drain or the like either.
The steep slope down into the side of the log house you posted about is a potential issue with your moisture management plans, depending on what (if any) drainage is built into/around the edges of what looks to be a driveway or road uphill (I'm guessing from the telephone pole). During a heavy downpour, if there is asphalt up there that means the water will mostly roll off and follow gravity due to the lack of permeability.
Did you notice any sort of possible "high water" marks inside the cellar walls? If there were any, that may have been part of why you were advised to keep the mechanicals upstairs. The fact that you described condensation dripping "from everything" is a bit concerning. Was it just on all the stone walls, or was the wood on the joists/ceiling wet as well?
It seems from the photos that there are some stepped exterior retaining walls for that slope, and I noticed the long distance of the base downspouts from the gutters. So it appears some remediation has been attempted in the past to help keep the water away from the cellar. The stepped retaining walls may have been an effort to break any rolling water from the possible paved surface above, allowing it several stages to pool before hitting the house foundation. Given the expense it would be, former owners likely did the stepped retainer rather than excavate on the sloped side(s) to install drainage, non permeable water barriers and insulation before refilling and creating the stepped retainers. Again, checking permit records will help you understand what was done prior for remediation.
I'd look into finding flood maps for the area as well, if you do research them do not discount the "hundred year flood" areas because that's a dice roll you don't want to make. I'm not sure where you are, but here in Massachusetts there is a wonderful mapping program called OLIVER that shows quite a bit of info about waterways, marsh, and flood zones of various year chances.
Also consider how much newer development uphill where the open grounds that formerly would have absorbed water in heavier rains are now less permeable. I have a friend down south that lives in a longstanding family property that never flooded until the town allowed development uphill from him, but let the development happen with no rainwater mitigation for the loss of formerly permeable natural land. Even after first digging swales, then installing French drains around all the uphill portions of the property, they are still having flooding problems even when the rains aren't prolonged downpour heavy. And from my understanding, the slope is much gentler than yours.
As to the concrete floors, what you would likely want to do is have a drainage ditch around the inside perimeter (usually gravel filled) that would lead to whatever outtake (sump, cistern). Your contractor will also probably put down an impermeable liner before pouring the concrete. If they have to dig down because the ceiling height is too low (not sure from images), then you'd need to have at least a quarter height foundation wall added inside the cellar's original foundation lines to keep the original from buckling inwards. In the latter case, I'm pretty sure one of our District members did something similar (but I admit I may have read it on another site), but heck if I could remember how to find the thread.
