Having more connections does not increase your chances of sprouting. Your chance of sprouting is directly related to the weight of beans in your wallet. The more beans the greater the probability of sprouting.
What having more connections does for you is to decrease the likelihood of 'generated but not accepted' sprouts that occur when you sprout at the same time as someone else on the network. In this case, it becomes a race to get your sprout broadcast to the most nodes on the network for adoption into the blockchain. The more connections you have the more nodes get the sprout transaction data directly from you instead of relying on other nodes to rebroadcast the data for you.
You do not have to run the computer 24/7, but the wallet cannot sprout if it is not connected to the network. Sprouts are the way the network rewards the user for contributing resources (processing transactions) to the network. If your computer must be shutdown half the time, your chance of sprouting will be zero during the downtime. During the time it is on, the chance of sprouting will be relative to the weight of your beans versus the total weight of beans sprouting on the network.