Does desktop wallet need to be online for the coins to mature? Suppose, I've installed a new coin wallet, got synchronized with the network, now I've sent some coins to my wallet. Now, if I close my wallet (and stay offline), would it get maturity after X amount of time (X is the coin maturity time) or would I have to keep my wallet online to get maturity ?
No, your wallet does not have to be online. Maturity depends on the block number in which a transaction was included into the blockchain.