Yes, it's possible to solo a lane with Kassadin. It's also possible to jungle with Sona, but I prefer not to.
Kassadin just doesn't have the speed, defense or mana management (even with a point or two in Nether Blade) to have the staying power against more common choices in 1v1 lanes.
Also Lestrang, you seem to be extremely fond of telling me what Kassadin can do to survive, and really clueless on what they can do back to you.
Do you think that Ashe cares that you threw a Q at her?
Do you think that Urgot cares at all that your E is charged up from his spamming?
They don't. Kassadin needs to be safe, alive and farmed in the early game. Give solo lanes to champions who can afford to take risks.