I was recently given my deceased father-in-law's service sidearm by his wife. Is it the property of the US Army? Can I legally possess it? Do i need to take some sort of action to legally possess it?
Yes you can legally possess it. Lots of guns made it to the private sector with the words US ARMY on it back when the government was nice enough to sell us surplus stuff.
I can almost bet for certain that if would be a mighty task to track the trail of that weapon and no one in the military would particularly care about it if its been a long time. Some of those mod 1911 are like a brother to some old soldiers and the idea of having to hand it back was inconceivable back then.
As far as possessing it, there is nothing that you need to do. In the state of Texas, there is no such thing as firearms registration, and no firearms registry, so there's nothing at all that you need to do.