Many of the weapons used in WWI were reused in WWII. Most not to the extent that they previously were, however. Many older weapons were brought back into service due to weapon shortages; the volksturm is a good example. The M1903 Springfield was still in use, the Gewehr 98s, Enfield rifles, M1911s, and the Luger P08 are a few off the top of my head.
Many. The Colt 1911 pistol and Springfield 1903 rifle (an infantry rifle in WWI, and a sniper rifle in WWII) come immediately to mind.
Yes, many weapons fought in both wars. This could turn into a list-post, so I'll try to elaborate on some of the more prominent weapons that were used in both conflicts.
As for the Russians, yes the mosin is the most obvious example. It's original form was the mosin 1891 long rifle, but a shortened carbine version was also made starting in 1907. Both fought in the first world war. The mosin 1891/30 was an overhauled version that was made from older rifles in 1930 and served as the main mosin variant in the second world war. Updated carbines were manufactured in 1938/44 respectively.
The American example would be the M1903 springfield bolt rifle. It was the standard issue rifle for American troops in the first world war, and was still issued to a few units stationed in the pacific at the start of the second world war, despite the growing prevalence of the M1 Garand. Its most notable role in the second world war was its use as a sniper rifle, as top-loading M1 made mounting a scope difficult.
The British counterpart was the Lee Enfield bolt action rifle. It had been around since the late 1800's and saw action with most of the British army throughout the first world war. Britain wanted to adopt a newer semi-automatic rifle in the years leading up to the second world war, but fearing that a new war would catch them unprepared, chose to stick with the trusty Lee Enfield. After some minor updating (Which led to the 4th generation of the rifle since it's creation) the rifle went on to see British troops through the second world war and even into the Korean war. Another notable British example was the Vickers water-cooled machine gun. During one encounter in the first world war, a company "fired their ten Vickers guns continuously for twelve hours. Using 100 new barrels, they fired a million rounds without a single failure."
There are dozens if not hundreds of other examples, but these are the most popular examples.
While they are wikipedia, these sources have great information regarding the history and development of these rifles, if you're interested.
http://en.wikipedia.org/wiki/Mosin%E2%80%93Nagant#Variants http://en.wikipedia.org/wiki/M1903_Springfield#Variants http://en.wikipedia.org/wiki/Lee-Enfield#Rifle_No._4
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