"Slab" is of slavic origin, but I wonder if there is some deep indo-european link to protogemanic "slap" meaning weak as in German "schlapp, schlaff", Dutch "slap". There seems however to be a connection to English "sleep" which translates to German "schlafen" and Dutch "slapen" (which is a weak, relaxed state)
There is, yes. Proto-Indo-European has *(s)leh₂b- which is related to the words you listed, and also to Latin labo, itself related to English labour (which in Latin could also refer to an affliction. Romance languages have lots of delightful origins for their word for work. Spanish trabajar comes from tripalium, a torture instrument with three stakes)
I think it’s kinda funny that the word for weak kinda sounds like “Slav” considering Romania’s location.
Well, this is actually a Slavic word. “Slab” also means weak in Serbian.