Not a full answer, but no US infantry units served in both theaters.
It wasn't uncommon for USAAF crews to fly a complete tour in Europe and then transfer to the Pacific theater, but this was on a man per man basis, not entire units. A rather famous example of this would be Paul Tibbets who flew 43 combat missions with the Eight Air Force in Europe before returning stateside to help develop the B-29. He then commanded the 509th Composite Group which conducted the bombings of Hiroshima and Nagasaki.
I don't know the answer to the 2nd part of your question, but as for the first bit:
The New Zealand 3rd Division fought the Japanese in the pacific theatre during 1943-1944, where it took part in the campaign in the Solomon Islands. In 1944 the New Zealand Government decided to disband the Division in order to free up manpower (which was needed for both agricultural work back home and to provide reinforcements to the NZ 2nd Division fighting in Italy), and therefore the Division was disbanded on the 20th of October, 1944. Around 4000 of its former members were sent to Italy to join the NZ 2nd Division.
So at least some men fought on both fronts, but not exactly as 1 unit.
The 2/1st battalion of the 6th division fought in both North Africa, Crete and Kokoda in the South Pacific. Veterans of both have said that the pacific was far worse. They suffered greatly in the hot humid conditions. With the large amount of rain malaria and dysentery were common. The other thing was in such thick jungle and tiny paths seeing more then a couple of metres in front of you was next to impossible.To give an idea of the conditions here is a photo of the track
https://www.awm.gov.au/collection/026852/.
Source australia goes to war 1939 to 1945 by Robertson J.
The 6th Marines were in Iceland and Guadalcanal, serving in both theaters. Of course there was no active fighting in Iceland.
A narrative style account of their time in iceland.
http://www.ibiblio.org/hyperwar/USMC/USMC-C-Iceland.html
The Australian 7th infantry division fought in North Africa and PNG.
http://en.wikipedia.org/wiki/7th_Division_(Australia)#North_Africa_and_Middle_East
Is it possible that any Soviet units fought in both theaters? I'm guessing it could be, considering Yang Kyoungjong fought in both theaters.
EDIT: Wikipedia article on the Battle of Moscow states:
Initially, the Soviet forces conducted a strategic defence of the Moscow Oblast by constructing three defensive belts, and deploying newly raised reserve armies as well as bringing troops from the Siberian and Far Eastern Military Districts.