The cleaning of any coin is not a good idea, but it also depends on why you think the coin needs to be cleaned, so take it to a dealer or collector and ask for an opinion
$2.00-$4.00 in average condition
Uh, the lady is Miss Liberty, as in "Statue of ..." Please see the Related Question for more.
It's a Roosevelt dime not a liberty dime and all are considered common, value is about $2.00
The Winged Liberty, or Mercury Dime. Most people call it the Mercury Dime, but its real name is the Winged Liberty dime.
It isn't a liberty dime, it is a Roosevelt dime, it isn't silver and only worth 10 cents.
A 1926 Mercury dime is common, average value is $3.00-$6.00 depending on the condition of the coin.
The Liberty dime minted in 1943 was 90% silver. The current rate for a circulated 1943 Liberty Dime is the current value of the silver of the coin, presently $5.50
You can find a picture of a 1936 liberty dime at the related link below.
That's off by about 65 years. Mercury dimes were minted from 1916 to 1945. A dime from 1853 would be called a Liberty dime or Liberty Seated dime.
A US dime dated 1867 is a Seated Liberty dime. To see one, in your search box type in Seated Liberty dime and click on images.
The Liberty Seated Dime (1837-1891)
The value of a 1842 seated liberty dime is about $20 in G-4(good) condition.