Short-term capital gains are typically taxed at your ordinary income tax rate, while long-term capital gains may have a reduced tax rate.

Would it be treated as short term capital gains or ordinary business income paid out as dividends from the LLC/S Corp? 

Make a run for real estate professional status to open the avenue to lowering your total AGI with real estate passive losses now being treated as ordinary losses.

I heard arguments that this is something called sweat equity and that the IRS will tax it as ordinary income.

However, it may not count towards your second material participation test for any more traditional rentals and get to 'work around' the passive activity loss rules - Short Term Rentals can be classified circumstance pending as "ordinary" type income, thus falls in a different bucket. 

Is depreciation looked at the same as ordinary business losses?  

This means you might be liable for capital gains tax on the appreciation in value that resulted from your efforts.Income Tax: If you're involved in a real estate project and are compensated for your time and skills, this compensation may be subject to ordinary income tax.

@Anthony RegaThe Gain and lending (providing a loan) on the home are two separate thingsYou will pay interest income (taxed at ordinary rates) on the interest you receive.

My accountant says that (to his knowledge) since my wife is a real estate professional, the gains from the sale would count as ordinary income and therefore could not be used to offset stock trading losses.I guess I'm looking for either a more-knowledgeable accountant, or a work-around.

If you qualify as a real estate professional, you may be able to deduct real estate losses against non-passive income, which can include ordinary income and potentially capital gains from real estate activities.However, the treatment of capital gains from the sale of stocks is generally separate from real estate activities.