1) will my SSRS 2014 reports run as is once the upgrade to SSRS 2019 has been completed or will I need to rewrite my reports?
I suppose the most reports will run. We had to test for layout changes, fonts , custom dll's, subscriptions, reports with oracle-statements ( change in interpreting comments ).
2) will there be a significant down-time for my users - time they will not be able to access current reports?
Try it out on a test environment. We had some downtime whilst migrating sharepoint 2012 to SSRS 2019 but that was including renaming reports, re-setup subscriptions.
SSRS 2019 database has to be 2012 or later. https://docs.microsoft.com/en-us/sql/reporting-services/install-windows/ssrs-report-server-create-a-report-server-database?view=sql-server-ver15
3) can SSRS 2014 be ran along side SSRS 2019 or is it best to just do the upgrade that way I'm only running one version?
Since 2017, only one instance of SSRS is allowed on the server (installs as instance SSRS)
4) will I need to also upgrade my Visual Studio software? If so, what version is best to use?
We mostly use the Report Builder, no experience with Visual Studio report deployments. I suppose VS 2019 would work best with SSRS 2019