How an LN node can be sure about existence / liveness of an individual path of channels?
Short answer: no (unless you control all nodes on that path)

However the problem about uncertainty is part of the routing anyway. You know the topology of the network with the cltv deltas, routingfees and capacity of ech channel. Yet you never know how the capacity is split into balance of a channel. so once your lightning nodes computed a potential path the onions are constructed and the routing process starts and will only be successful if all channels on the way have enough satoshis on their side to forward the payment. The next time this path is supposed to be used again they have to construct the onion and get feedback from the nodes on the path to see if the payment can actually be routed through. 

So to some degree it seems to be of little use to store previous paths that have been used. 

if however channels would be closed or new capacity is spliced in (once splicing is implemented) nodes would get this information via the gossip protocol.