Relationship is not explicit. fitness for a block is just total number of endorsements in the chain that block is extending. If you are well connected with endorsers and bakers, fitness is very likely high. Answer that you are looking for is empirical which requires logging number of peers connected vs fitness score of the baked block for each baker.
Connection to 20 or 10 or 5 good peers vs any peers will also make a difference. Right now p2p layer tries to keep a set number of connections with good peers (configured by user). Goodness definition used by p2p management(right now kicks out bad or inactive actors who send invalid block and transactions) might not directly impact your network reachability (It is an interesting problem to solve) and thus not have any impact on fitness.
EDIT: to give context, we have seen in the past some instances where priority 0 (P0) bakers do bake but their block gets taken over by P1 bakers due to lower fitness because they contained less endorsements than the P1 baker did. So this could be due to not enough endorsements seen by the P0 baker.
Yes. This might be one of the reasons. It could be also the case where P1's block reached more endorsers before P0's did (endorser can't wait forever for higher priority block as his reward is proportional time when it signs it, signs the 1st valid block they see). So endorsers signed endorsement for P1. As endorsement for a block are included in block that extends it, A block's fate can depend on when it reaches endorsers and also on whether endorsement operations reach next block proposer.