Unless you have your stuff really dialed in; separate bank accounts, meetings, minutes, by-laws, resolutions for everything you do and you end up in court, it will be very easy for opposing council to pierce the corporate veil and hold you liable.

We will use, say, the $40k in our LLC bank account to make the purchase (we want the purchase to be made by the LLC to avoid any issues with piercing the veil, etc) to buy and hold, rehab, and eventually rent out--plot twist: We want to live in the home for the first year or so then move and rent it out.

They then take your property, your primary residence, garnish your wages, etc. to attempt to recover the additional 2M.Under the LLC, your downside risk is substantially minimized (assuming you don't pierce the corporate veil). 

This will NOT be the first time you will deal with this with this tenant so save yourself some agony and get rid of her now.

Even if the property is in an LLC or corportation, there is always the possibility that the claimant could pierce the veil of the LLC/corp.   

But remember that LLC give you "limited" protection, there are still ways to pierce that if someone really wants it.

But one quick way for a lawyer to pierce that LLC veil is to state you did not operate your business in a proper manner, and going around work comp law would be pretty good proof that you did not operate your business legally.  

You want rentals in different LLCs because, if one LLC gets sued, the asset on other LLC won't get touched (given if you there is no piercing of corporate veil).

The reason stated by RyanFor a very similar but subtly different argument, It could be argued that the buyer or seller did not know they were dealing with a LLC, therefore the legal liability is pierced. 

As long as you don't pay for personal items out of the business entity or pay for business expenses personally, it's much more difficult to pierce the corporate veil.