How much should I aim to have saved up for down payment, expenses, etc?

IF you can supplement or eliminate your living expenses all together, your saving rate will go through the roof allowing you to buy that much more property. 

My recommendation is to find something where you can rent 1/3 or 1/2 of it right away so that you can supplement your income or decrease your negative cash flow to make your savings last the longest. 

She said it may be a couple weeks before she’s ready with all of the supplements and such with her insurance company.

Are you looking for monthly income to supplement your earned income?

Especially because I am aiming at $0 down 

We are aiming to BRRRR our next property potentially, or find a 10+ unit and the capital they are offering is looking tempting as our investment accounts are getting thinner.

Helping to place loans that can't fit conventional financing could help supplement your income.

Why not suggest she buys her own window AC units to supplement?  

I am getting a feeling this would be a mistake trying to rely on airbnb to possibly supplement my mortgage with how the economy looks.