I current run
https://apicoin.io, and we have push tx functionality (
https://apicoin.io/api/v1/doc/#pushtx). We are also the currently the only ones that can do it in batch, up to 100 signed transactions in one api call. We use special technology that we designed to do that, so you aren't going to load up any premade software to do that.
Are you dreaming?

Minimum 4 GB per coin and perhaps more. And use load balancing

I would assume we could cope with a DigitalOcean $10/month (1GB RAM, 1 Core, 30GB Disk, 2TB Transfer)
Also this impossible, you would need at least 75GB. But then I wouldn't also use that same VPS for the frontend otherwise you will be running very slow.