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 matter 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.
If by "confirmed" you mean another block follows it, then yes. The block that is first to have the
next block built on top of it wins. Of course, the earlier block has a head start, but it is not guaranteed because the chain follows the longest branch, and the later branch could get lucky. And, if the next blocks are produced at about the same time, then it is the
next block that determines the winning branch. And so on ...
I am solo mining with my own blockchain and wallet hosted by me, I have opened the firewall to allow inbound connections, not just outbound, I figured it may be relevant in the off chance of success.