We send out two rate limit messages: one for API calls, and one for feed updates. Our API call rate limit is about 2000 activities per minute, but feed updates are more like 50-100 per minute on the free plan. Setting up a follow relationship will trigger some feed updates as old activities get copied from other feeds to the new follower's feed.
When you do hit a rate limit, we don't stop your incoming traffic, but we de-prioritize slightly it so it takes a little longer to catch up. Our API v2 coming out soon will report rate limit information in API calls so you'll have more visibility into how close you are to hitting those limits before getting emails.
Regarding timeouts, which region is your app in (us-east, us-west, eu-central) and where are you located related to that area? We're going to be rolling out multi-region support later this year to minimize latencies there as well.