While in a normal conversation(informal), would it be natural to omit things like "ho", "sto", etc while using the verb forms that correspond with them(I don't know what they're called)? Or would this just be straight up bad Italian?

See a translation

In italian you can’t omit “ho” and “sto” because they are verbs.
“Ho” = I have.
“Sono”, “sto” = I am, I stay.
It depends on the context.. used “sono” with nouns and adjectives, and to express that you are in a place in that moment; used “sto” to express your temporary physical condition, to express that you are available for something).
You can also use “sto” to express the present continuous. For examples: I am studying = Io STO studiando; I am eating = Io STO mangiando.

I know it isn’t simple and I hope to have been clear😊