What we do:
ShowMax back-end is in-house developed micro-service oriented platform. We do embrace a lot of open-source projects as well. As we use OSS we like to give back as well through reporting bug and/or committing patches.
Majority of our back-end services are written in Ruby or Go. We use EventMachine when writing apps in Ruby (Goliath as application server, executed via EinHorn). Go is typically used in apps, which require large throughput and are simpler. Example of such can be end-device event collection or routing of storage requests. From the open source projects perspective, you can meet ElasticSearch, PostgreSQL, Varnish, HAProxy, nginx, HBase, heka, RabbitMQ, Kafka and many others.
Platform is not only about APIs. Primary business is streaming videos, to be able to do that, we have own encoding pipeline which can orchestrate encoding on hundreds of servers. We have developed own request routing layer for our origin storages to be able to scale throughput and storage capacity as we need.
Planning and housekeeping of tasks is done via Phabricator (task tracker spin off Facebook). All our code is in git (we have own installation of GitLab). We use Kanban style of development process. As usual, we would use tools best fitting the job, despite being open-source, free or commercial.