That's pretty much what happens but no amount of routing can make up for the loss of capacity when you physically take out a part of the network. You end up with less total available bandwidth and more latency.
It is, and a cut like this rarely causes a complete outage, but it depends on whether and what kind of backup connectivity your ISP has, and the connectivity of your destination. Bigger entities tend to have more backups. Smaller entities tend to have fewer backups, and often the backups are much lower bandwidth/quality, so it's typical for service to degrade.