Money is not fundamental to capitalism. Capitalism could exist just fine without money. Money (according to Marx) is a commodity just like any other commodity. The only difference is that many commodities have a use-value that would still exist outside of capitalism. (food can still be eaten, shoes could still be worn etc.) Money on the other hand has as it's only use-value that it lubricates the process of exchange. Money does not have a stable exchange value because it is used as a unit to measure the exchange value of other commodities.
When you go through a process of exchange to get money, money is not actually the thing that you want, but rather, you see money as it's potential to quickly and easily exchange itself for other commodities.
To 'abolish money' would make capitalism slower and more cumbersome. (The minute I see "Make Capitalism Slower and more Cumbersome! (A)!" on a banner I will lower my black flag and burn my (A) card.)
There are places in the world where money is not the most efficient commodity at lubricating exchange. In a sense money "losses" its use-value. In places where the exchange value of money tends to go down quickly or change rapidly people often will use a different commodity to store their wealth in. This other commodity can become a 'new money' even though it has an alternate use value and was not designed for this purpose. (Brick is a common 'new money')(I wish this were a settlers of cattan joke, but it's real shit.)
As long as commodities exist, there will be a commodity that performs the task of money.
A better known and more humorous story about money is when money gains additional use-values. There have been times and places where money was not properly performing its function of lubricating exchange because it's exchange value was so low. (We have all heard stories about pre-WWII Germany where it took a wheelbarrow full of money to buy a loaf of bread.) At these times and places people find new uses for their gigantic stacks of useless money such as, wall paper, kindiling, or even fuel for a fire for heat or cooking purposes.
The main critics I can see of 'Money' are the Zeitgeist Movement wingnuts. They argue instead for a "resource based economy" which is an entirely meaningless term. To refer to the world as 'resources' already shows a degree of alienation and strongly hints towards a replication of the commodity-form.
To answer your second question, no. Abolishing money does nothing to end the State, end Capitalism, end Prison, Borders, Imperialist War, Wealth/Poverty, White Supremacist Settler-Colonialism, Cis-Hetero-Patriarchy, Alienation, Work, etc., etc., or any of the constant abuses that come from these things.
For more on Money and Commodities read: Capital by Karl Marx