"Came out of the gates" is an expression from horse racing. It basically means STARTED OFF.

She enthusiastically embraced the stay-at-home order, and was full of good resolutions about what she would do with her time.

It seems the author means this thing got in the way of her busy routine, yes. And as for “coming out of the gates pretty darn hot”, it would probably be useful to have a bit more context, but it could refer to being excited, riled up, “in action”, you could say. For example, it’s often employed in police/army shooting scenarios in movies. In general, you could say it means “ready to engage”, which can be adapted to a number of contexts. If you’d like to have a chat, send me a message and I’ll be glad to help you broaden your vocabulary through some stimulating exchanges!

Okay, I did. I think, considering the context being “a busy work routine interrupting and forcing a person to get new habits”, “coming out of the gates pretty darn hot” would mean that at that point, when this thing happened, the speaker was used by the overwhelming work habits she had. Kinda like “heated up by work”, in a very literal way, I guess. Imagine a device that overheats due to excessive use, it’s likely the meaning she’s trying to convey!

Sure