There is not a way to automatically launch iExplorer when you connect your iPhone, but you can close the iExplorer window and have it still run in the background. It also has a preference to launch in the background when you login to your OS user.
As for being a replacement for iTunes, iExplorer does share some of the same features of iTunes and also lets you do a lot of things that iTunes doesn’t let you do, but it should not be considered as a total replacement to iTunes since it’s mostly designed to transfer iTunes media and other data to a computer.
As for installing apps on a device, you’ll need to use iTunes or the App Store for that, but iExplorer does let you transfer app data between your computer and iPhone such as game saves, documents, and other personal app data.
These articles explain a little more about it: