If you are sure the outside drain is the culprit, I suggest you contact a professional waterproofing contractor to either re-route the drain or install a sump pump for your protection. This is nothing to mess around with, and it is only going to get worse over time and lead to bigger and costlier problems.

Having over 30 years experience in basement waterproofing, my best advice is to slope the floor towards the drain to avoid any possible issues.

Believe it or not, any flooring is possible in a basement- but, as long as the basement remains dry. Once there is moisture intrusion, any type of flooring is susceptible to damage. I suggest waterproofing the entire cove area before installing any type of flooring.