Does the number of inbound and outbound network connections to your blockchain play any role in who is awarded the block? For example, if a solo miner solves a block and then say 1 second later a large pool solves the same block before all the nodes record the block in the blockchain? Who wins?
That's called an Orphan race. Which found block is confirmed first wins. So, yes, connections matters but not how exactly many you have but rather how fast of a connection they have to the Bitcoin network so their found block is available to be confirmed.