Proxy servers are a bit like Bitcoin wallets - They're only safe if you've set them up yourself.
Proxies are still very common on the web any many people use them to bypass geographic restrictions of streaming portals like Netflix.
They also hide your real IP address for your target (not for the proxy itself) and make you somewhat anonymous.
Yes! In fact you can create your own JS infecting proxy in a few simple steps with a Linux server and Open Source tools. Here is a simple tutorial: Why are free proxies free? (2013 - blog.haschek.at)
This technique also works with https if the site loads unsafe resources (eg. jquery from a http site).