Deductions & credits

@Rk99 Glad to help.

Hmm... I'm confused. Do you own 12.5% of the apartment? Or 12.5% of some land, a house, and the apartment? For the rest I will assume that we are only talking about the apartment and that your mom owns 50% of it and 12.5% x 4 for the kids.

A couple of things.

Right now you own 12.5% of the property. Your holding period (for > 1 yr capital gain) started when you inherited it. Your basis actually appears to be its fair-market value as of that date. I say appears to be because it is somewhat surprising that it applies if your Dad wasn't a US citizen or resident. But it appears to. Please review this for more info https://hodgen.com/basis-step-up-on-assets-inherited-from-nonresident/

You mother will be gifting you (as I understand) another 12.5%. So you will own a total of 25%. Your basis in the gifted 12.5% from you mom is her basis. (You are right that your gain is long term is she owned it for > 1 yr. Living it in does not matter at all here.)

So you are not selling the apartment. You are selling 25% of the apartment. That is what I would put down in TT next year. (E.g. "25% of apartment, 1 main st, town, state, county")

Your total basis is therefore the 12.5% of the fair-market value of the property when you inherited it plus your mother's basis in the second 12.5% that she is giving you. 

Her basis in her 50% of the apartment is 50% of what she and your father bought the property for plus any improvements. 

Let's say the property was worth $100k when your father died and that your parents total cost for the apartment was $40k. Then your inherited 12.5% basis would be $12.5k. Your mom's 50% basis would be $20k. Therefore your total basis would be $12.5k + $20k / 4 = $17.5k.

As you say getting the records maybe difficult. In the US I would suggest pulling the recorded deed to see the price paid. In any case the burden of proof is on you. The IRS will be quite happy if you say the basis is zero. That might not be far off for a 50 year old property. So it may not be worth worrying about.

Similarly you need some sort of appraisal or tax document or something to establish the fair-market value when you inherited your $12.5k. Again it might not matter if you low ball it if you are either in a low tax bracket, the gain isn't much (there is a zero % LTCG bracket for some). Or if the  Italian capital gain tax is more than the US tax and the foreign tax credit magic works for you to offset the US tax. The latter is likely, but not certain.

Also note that you need to convert to US dollars. This describes how to do that. If there are only a few dates involved you should use the exchange rate on those dates. If there are many, you can use the average annual exchange rate.

https://www.irs.gov/individuals/international-taxpayers/foreign-currency-and-currency-exchange-rates