How sick was Franklin D. Roosevelt? And how aware was the American public of FDR’s physical trouble?

To your first question, I've previously written in detail about FDR's health in 1944 and 1945; in summary, he was extremely ill and people close to him were quite concerned at his deterioration.

As FDR refused to discuss either his condition or prognosis with even his physicians, nobody besides them really knew how sick he was - but there was enough worry among party leaders just based off of FDR's appearance and interaction in meetings that they succeeded in pushing through Harry Truman's nomination for Vice President in 1944 with the eye on the distinct possibility that FDR might not make it through his fourth term.

Even Truman himself, though, didn't want to face up to reality. While he witnessed FDR's hands shaking badly to slosh the drinks during their sole one-on-one luncheon in 1944 and wrote to a friend afterwards about the incident and his shock at just how bad FDR looked, afterwards he proceeded over the next six months to pretty much ignore the implications of what FDR's health meant to him.

As far as the American public, there are a couple incidents worth noting. First was FDR's well publicized campaign appearance at Yankee Stadium shortly before the 1944 election where FDR drove up to the platform in his convertible, sat in it while giving a rousing speech sounding much like he had in 1932, and most whisperings about his health got tamped down enough so that it wasn't an issue by Election Day. Second, though, was in the spring of 1945 when he returned from Yalta and addressed a joint session of Congress. Not only did FDR give the speech while sitting down, but explained for the first and only time in his public life why he was doing so: that he was carrying around 10 pounds of steel on his legs. This was to be the only acknowledgment of his disability, and while it didn't raise that much of a ruckus especially when he immediately followed it up by noting he'd just completed a 14,000 mile trip, in retrospect it was an amazing admission.

As far as world leaders reacting to his death, both Stalin and Churchill knew FDR was sick but much like the rest of the world at the time of his death were genuinely surprised that he was that sick. (Stalin was mostly disappointed since he thought he had then gotten to the point where he could get the best of their interactions; Churchill seemed to actually grieve.) Keep in mind, though, both of those leaders had their own health issues; Churchill's first heart attack was mostly kept from him, but Stalin's May 1945 one was significant enough so that it meant that some of the Yalta preparations changed.