Do you want to collect your ESXi logs?
You can use "Syslog Collector" on this regard.
There is no recommendation about logs retention.
For the snapshot question. You can use PowerCli to find out if the snapshots were quiesced or not.
You can run the command below. There are also many variations of code you can use to get different types of reports...
Get-vm | Get-snapshot | fl
The Quiesced attribute will show either True or False.