There's no warmup time. We spent some time with various OSS FaaS tools and that was pretty killer. When you deploy, we build a v8 snapshot of your code, then push it out to all the Edge Servers. It's already in memory and ready to rock when a request comes in — less than 1ms to get to your code for most apps.
fly.cache does persist, although it will vary per region (that's a surprising thing we learned about CDNs, cache contents vary per region quite a lot). It's volatile so you shouldn't rely on it for durable data storage, but you also pay for what you use so we have a strong incentive to not evict cache data any more than you request. :)
The global context _can_ be shared between requests but you can't really count on it. There's no guaranty that one request will run in the same context (or isolate or server) as a previous one.
We have some ideas for more durable and structured storage down the road. Building APIs that are location agnostic is fun, but a little mind bending.