To your first question, it's called a Treasure Goblin. You chase him, usually through other enemies, and he drops items and stuff when/if you kill him. To your other question, yeah pretty much
Do you have a puzzle ring equipped? The affix on that means a treasure goblin follows you around picking up junk and common items. After it picks up enough it will drop a rare item that has a chance to be legendary
I assume you are still levelling, once you replace it, hang on to it. You can transmute it in Kanais Cube (just it on its own) to open a portal to Greed's Realm where riches await!
Level 49 right now! On act 3. I’ve been saving gold and right now I have about 400,000. I haven’t found any reason to spend a large portion besides repairs and such. Is it save to just do what I’m doing and save most?
Absolutely! You don't really need much whilst levelling, but it gets expensive quick at max level, as gold is required for things like levelling crafters, combining gems and enchanting gear.
Fear not though, gold drops pretty readily and there are ways of getting a nice chunk, such as visiting Greed's Realm. You may even get a treasure goblin to drop a portal randomly on kill.