OP here!
So I've been able to create an answer based on some research:
#Get report folder path
$ReportPath = "C:\Users\USER\Downloads\*\Report"
$MasterReportPath = "C:\Users\USER\Downloads\MasterReports"
#Rename report folder to {currentparentname}report
Get-Item -Path $ReportPath | ForEach-Object {$a = $_.FullName | split-path -Parent | split-path -leaf; Rename-Item -Path $_.FullName -NewName $a"Report"}
#Move report folder
$AnyNamedReportFolder = Get-Item "C:\Users\USER\Downloads\*\*Report*" -Exclude *.jmx, *.csv
Move-Item -Path $AnyNamedReportFolder -Destination $MasterReportPath
Definitely isn't elegant, but does the job. Since I have the main answer to this question, I'll mark it as answered. However there is an issue with this, which is that if this is run multiple times then same named folders will not move (since it doesn't append a unique number or character). I've highlighted this in a new question here, should you be interested. If all you need is a one time working script, then the above script worked for me.
I've now also answered my secondary question, but for the sake of not cluttering this question, please refer to the link under my answer. Thanks!