How to assist target with a controller in FFXIV

In the Store page of every Steam Game you can check if one of these symbols is available.

or

Usually in all recent games (2-3 years), if a controller is enabled, there are high possibility that Microsoft Xbox 360 works and is well supported.

But to be sure I always suggest to take a look to that game specific forum. Most of times there are users that already asked for Xbox 360 compatibility.

The aim assist is available only in the campaign not in the multiplayer. The way it works is that it helps you 'snap' to the target nearest to the reticle while aiming down the sights. So maybe that snapping action is throwing you off. You can disable it and try.

It takes time to transition from PC to controller and while keyboard and mouse maybe more accurate, you will get used to the controller soon enough.

Now to answer the other part of your question, there are some non-standard controllers (read cheats) that provide extra functionality like this one. But you can't use a keyboard and mouse to play regularly.