@USM Professor
I will have to think about your figure more specifically, but in general from this response:
1. There was no will. If under operation of Arizona law, each sibling inherited 1/3 of the home, that is what happened.
2. Any support you provided before your father's death is money you freely paid out, and there is no legal way to get recompense from your siblings. (They can offer to make an adjustment, of course, but there is no way to compel them, and that money plays no factor in determining the tax situation of the home.) Costs you paid after your father died should be reimbursed to you before the estate pays any beneficiaries (before the estate divides any other assets, such as a car or bank accounts.)
3. Under the circumstances, the $26K offer could be used as proof of FMV as of the date of death, but if it would be advantageous to claim a higher value, you could probably use the other circumstances of the buyer as proof that the offer was not really a fair market offer (as long as you also had some backup for claiming a higher value).
4. There is a difference between repairs and improvements. Repairs do not adjust the basis. Repairs maintain the property in as-is or as-was condition. Improvements, sometimes called "betterments", increase the value of the home or extend the useful life of the home or one of its systems. Painting is normally a repair, as would be fixing a leaky faucet. Improvements include things like replacing windows, doors, carpet, air conditioning, remodeling a bathroom, and so on. (However, if you remodel the entire kitchen, which includes painting, that painting would be included in the cost of the improvement, even though painting other rooms might be considered repairs. See publication 523.
https://www.irs.gov/pub/irs-pdf/p523.pdf
5. Some selling costs also increase basis. See publication 523.
6. Ignoring all other factors for the moment and assuming the entire $31,000 was "improvements", each sibling inherited 1/3 of the house worth $8,833. Your siblings gave you their shares of the house for a $3,000 payment and a gift of $5,833. Then you paid $31,000 for renovations. So the starting value for your basis is $45,833.
7. You may need to reduce that basis if some of your costs would be considered repairs instead of improvements.
8. Normally, when you give someone a gift, you also give them your basis in the gift. For example, if I give you a baseball card that cost me 10 cents as a child, and you sell it for $100, your basis is 10 cents, and the remaining $99.90 is capital gains. That suggests that even though you paid $3000 for each share of the home, you also received $5,833 in cost basis from the gift portion. That would raise your overall basis when you sold the home. However, this would require a bit more research on my part, due to the fact that the givers were related to you, and no gift tax was paid.
9. You never said what you sold the house for. Remember that you can include in the costs of selling, the real estate commission (if you paid one) and certain taxes and fees if it is customary for the seller to pay fees in your area. For example, the seller might pay a transfer tax to the county, or a fee to record the deed, or it might be customary for the seller to pay for a radon inspection. Essentially, you can add to your basis (or reduce your selling price) any cost that the seller must pay, that the seller would pay regardless of whether it was a cash sale or a mortgage. Costs specifically required to get a mortgage (or allow the buyer to get a mortgage) are not adjustments to the selling price. See publication 523.
If you go through the numbers again, and need to think more about your adjustments or have more questions, let us know.
