Variable scope within foreach-object in Powershell

1. How is Write-Host $letters not using the assigned variable $letters?

Because it is outside the ForEach-Object loop, hence, you're only reading the last result from that variable assignment. You probably wanted:

$thisDataArray | Select-String -AllMatches -Pattern $regex -CaseSensitive | ForEach-Object {
    $letters = $_.Matches.Value
    Write-Host $letters
}

Though, is worth noting that output to console is implicit unless captured or redirected, so, Write-Host might not be needed nor is a variable assignment:

$thisDataArray | Select-String -AllMatches -Pattern $regex -CaseSensitive | ForEach-Object {
    $_.Matches.Value
}

2. Why do I only get the one array member 'trouble' from my $regex?

You don't, you would get cadabra and trouble assuming Write-Host is inside the loop.

3. And finally. Why are there spaces between the chars of trouble e.g t r o u b l e

Because your regex [a-z] matches a single character and since -AllMatches is being used, all appeareances of a character match are being outputted. The reason why they then appear concatenated by a space is because Write-Host is stringifying the $letters array and joining it by $OFS (default $OFS is a space). i.e.:

PS ..\pwsh> $letters      
t
r
o
u
b
l
e

PS ..\pwsh> Write-Host $letters
t r o u b l e

PS ..\pwsh> [string] $letters
t r o u b l e

PS ..\pwsh> $ofs = ''
PS ..\pwsh> [string] $letters
trouble

• If you want to match all letters from a to z you could use:

$regex = "[a-z]+"

• If you want to match those elements $thisDataArray which have only letters from a to z then you could use:

$regex = "^[a-z]+$"

As for the "error" The variable $letters is assigned but never used, I assume you're referring to a PSScriptAnalyzer Rule, namely UseDeclaredVarsMoreThanAssignments, which simply means, that the variable being assigned inside a script block (ForEach-Object -Process { ... } in this case) is never used inside that script block: