At Cloudflare, we're working on expanding Workers (https://www.cloudflare.com/products/cloudflare-workers/) to allow access to your existing DB servers & offer protection with Argo Tunnel (https://www.cloudflare.com/products/argo-tunnel/). We are also enabling Workers to write into Cloudflare’s globally distributed cache, reducing retrieval time for repeated query results. We hope this will be a differentiator with using Cloudflare & highly valuable for your use cases.
Workers don't run Docker containers intentionally. The goal with Workers is to run with a lower memory overhead (~3 MB) and lower startup time (~5 ms) than you can get with full container isolation. This allows your Worker to run affordably in 150+ locations around the world. In many ways it's the ultimate destination for serverless, running code in a multitenant process where all you manage is your code.