Can you mine smaller cryptocurrencies - e.g. coins found in an ICO etc? E.g. any of the coins found here:

https://coinmarketcap.com/
"Can you mine smaller cryptocurrencies?" 

if they are mineable: obvious yes. (otherwise they can not be used to execute transactions so they would not have any benefit.)
but there are also some not mineable cryptocurrencies, for example ripple or cryptocurrencies which have 100% proof of stake.

"... e.g. coins found in an ICO etc?"

nearly all ICOs use tokens. a token is not a mineable coin. a token exist due to a smartcontract of another cryptocurrency. so the creator of the smartcontract can decide who gets how many tokens.