I really with they would add an init() callback that is called on cold start but before any traffic is sent to your lambda. It wouldn't help when there are no lambdas running but it could be useful when things are scaling up, especially if you can ask for additional concurrency above the actual concurrency necessary for spikes.
I don't think so. When it spins it up, the request is already in flight. Otherwise this would have been solved by everyone but instead everyone sees terrible cold start times.