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Quick performance question
Does the database performance or the query processing time by any chance depend on the memory you are assigning to the VM on which Vertica runs? May be a silly question for many of you, but I don't know the answer. 
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Your question is a broad topic that covers many key parts of Vertica. My best understanding is, surprisingly, memory is not as important as you might think. Very consistently the experts I've spoken with report database performance is tied to the correct configuration of resource pools. Take a look at some past user comments:
https://forum.vertica.com/discussion/236836/usage-of-resource-pools
Then take a look at:
https://my.vertica.com/blog/a-method-for-vertica-workload-classificationba-p223339/
Statistics are a key part of picking an optimal query plan, thus affecting query performance and to a certain degree database performance.
https://my.vertica.com/docs/8.1.x/HTML/index.htm#Authoring/AdministratorsGuide/Statistics/CollectingDatabaseStatistics.htm?TocPath=Administrator's%20Guide|Collecting%20Database%20Statistics|_____0
There is also a the concept of query performance tuning:
https://my.vertica.com/kb/Troubleshooting-Vertica-Query-Performance-with-System-Tables/Content/BestPractices/Troubleshooting-Vertica-Query-Performance-with-System-Tables.htm
Resource pool configuration affects the pool's "query budget" which directly impacts query performance. Too small of a query budget causes queries to single-thread rather than utilize all cores.