You’re over thinking a lot.
Every other downpipe clearly empties onto the pavement - yours appears to be the odd one out? I’d lift that metal cover - you may well find its emptying into the soil below the pavement given there won’t be a “drain” for you alone - if so its your down pipe and someone in the past had a bright idea… if so the rainwater will just hit the soil and disperse one way or another
Every house has some damp at street level - think you’ll probably find water just pools near the wall. Every house looks like the next as far as covering/lichen/damp looking - so … sort of how they were built
It is a cellar … living space is above the damp line. It was built that way - as long as its a bit damp (as opposed to wet through and crumbling) and not showing in the living space I’d be accepting it rather than fighting it beyond ventilate the cellar. Can’t make out what the grill/window do - but they’re either stopping rain entering the cellar or stopping it breathing.
And its a part time home - erm … it needs air and heat and living in.
Either way it looks a bit mountains out of molehills - look in July … it’ll be dry probably
Not that there aren’t solutions - but the street isn’t yours so French drains aren’t an option and I doubt the local commune will rely it with correct slopes for a holiday homes mildly damp cellar (that’s a polite of saying they’ll go Les Anglais and raise their eyebrows). Damp isn’t strange - we have a weird attitude to it
The cellar can be at or below water line - mid winter mine gets a bit damp - mid summer it dries out - that looks early 20th Century - maybe mid-late 19th Century - they didn’t create hermetic boxes then
But the simple answer after you’ve paid endless experts - is put the drainpipe back where it should be and accept it spews out onto the pavement and then accept cellars aren’t dry - but ventilation always helps - but you do need a lip on the cellar window (two bricks high roughly) so rain doesn’t just pour in if it bothers you. Equally remember its survived a 100 years …