"Run-time error 462 : The remote server machine does not exist or is unavailable" when running VBA code a second time

First problem : Run-time error '462' : The remote server machine does not exist or is unavailable.

The issue here is the use of :

1. Late Biding : Dim Smthg As Object or
2. Implicit references : Dim Smthg As Range instead of
   Dim Smthg As Excel.Range or Dim Smthg As Word.Range

So you need to fully qualified all the variables that you set (I've done that in your code)

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Second problem

You work with multiple instances of Word and you only need one to handle multiple documents.

So instead of creating a new one each time with :

Set WordApp = CreateObject("Word.Application")

You can get an open instance (if there is one) or create one with that code :

On Error Resume Next
Set WordApp = GetObject(, "Word.Application")
If Err.Number > 0 Then Set WordApp = CreateObject("Word.Application")
On Error GoTo 0

And once you've put this at the start of your proc, you can use this instance until the end of the proc and before the end, quit it to avoid having multiple instances running.

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Here is your code reviewed and cleaned, take a look :

Sub Docs()

Dim WordApp As Word.Application
Dim WordDoc As Word.Document

' Control if folder exists, if not create folder
If Len(Dir("F:\documents\" & Year(Date), vbDirectory)) = 0 Then MkDir "F:\documents\" & Year(Date)

' Get or Create a Word Instance
On Error Resume Next
Set WordApp = GetObject(, "Word.Application")
If Err.Number > 0 Then Set WordApp = CreateObject("Word.Application")
On Error GoTo 0

Workbooks("exampleworkbook.xlsm").Sheets("examplesheet").Range("A1:C33").Copy

With WordApp
    .Visible = True
    .Activate
    Set WordDoc = .Documents.Add
    .Selection.PasteSpecial Link:=False, DataType:=wdPasteRTF, _
                Placement:=wdInLine, DisplayAsIcon:=False
End With

With Application
    .Wait (Now + TimeValue("0:00:02"))
    .CutCopyMode = False
End With

With WordDoc
    .PageSetup.TopMargin = WordApp.CentimetersToPoints(1.4)
    .PageSetup.LeftMargin = WordApp.CentimetersToPoints(1.5)
    .PageSetup.BottomMargin = WordApp.CentimetersToPoints(1.5)
    .SaveAs "F:\documents\" & Year(Date) & "\examplename " & Format(Now, "YYYYMMDD") & ".docx"
    .Close
End With

' export sheet 2 to Word
Workbooks("exampleworkbook.xlsm").Sheets("examplesheet2").Range("A1:C33").Copy

Set WordDoc = WordApp.Documents.Add
WordApp.Selection.PasteSpecial Link:=False, DataType:=wdPasteRTF, _
                        Placement:=wdInLine, DisplayAsIcon:=False
Application.Wait (Now + TimeValue("0:00:02"))

With WordDoc
    .PageSetup.LeftMargin = WordApp.CentimetersToPoints(1.5)
    .PageSetup.TopMargin = WordApp.CentimetersToPoints(1.4)
    .PageSetup.BottomMargin = WordApp.CentimetersToPoints(1.5)
    .SaveAs "F:\files\" & Year(Date) & "\name" & Format(Now, "YYYYMMDD") & ".docx"
    .Close
End With

Application.CutCopyMode = False
WordApp.Quit
Set WordDoc = Nothing
Set WordApp = Nothing

' Variables Outlook
Dim objOutlook As Outlook.Application
Dim objMail As Outlook.MailItem
Dim rngTo As Excel.Range
Dim rngCc As Excel.Range
Dim rngSubject As Excel.Range
Dim rngBody As Excel.Range
Dim rngAttach1 As Excel.Range
Dim rngAttach2 As Excel.Range
Dim numSend As Integer


On Error Resume Next
Set objOutlook = GetObject(, "Outlook.Application")
If Err.Number > 0 Then Set objOutlook = CreateObject("Outlook.Application")
On Error GoTo 0


Set objMail = objOutlook.CreateItem(0)

' Outlook
On Error GoTo handleError

With Sheets("Mail")
    Set rngTo = .Range("B11")
    Set rngCc = .Range("B12")
    Set rngSubject = .Range("B13")
    Set rngBody = .Range("B14")
    Set rngAttach1 = .Range("B15")
    Set rngAttach2 = .Range("B16")
End With

With objMail
    .To = rngTo.Value
    .Subject = rngSubject.Value
    .CC = rngCc.Value
    '.Body = rngBody.Value
    .Body = "Hi," & _
            vbNewLine & vbNewLine & _
            rngBody.Value & _
            vbNewLine & vbNewLine & _
            "Kind regards,"
    .Attachments.Add rngAttach1.Value
    .Attachments.Add rngAttach2.Value
    .Display
     Application.Wait (Now + TimeValue("0:00:01"))
     Application.SendKeys "%s"
  ' .Send       ' Instead of .Display, you can use .Send to send the email _
                or .Save to save a copy in the drafts folder
End With

numSend = numSend + 1

GoTo skipError

handleError:
numErr = numErr + 1
oFile.WriteLine "*** ERROR *** Email for account" & broker & " not sent. Error: " & Err.Number & " " & Err.Description
skipError:

On Error GoTo 0

MsgBox "Sent emails: " & numSend & vbNewLine & "Number of errors: " & numErr, vbOKOnly + vbInformation, "Operation finished"

GoTo endProgram

cancelProgram:
MsgBox "No mails were sent.", vbOKOnly + vbExclamation, "Operation cancelled"

endProgram:
Set objOutlook = Nothing
Set objMail = Nothing
Set rngTo = Nothing
Set rngSubject = Nothing
Set rngBody = Nothing
Set rngAttach1 = Nothing
Set rngAttach2 = Nothing

End Sub

If this is running in Excel then you probably need to specify that CentimetersToPoints is coming from the Word library. As it stands, VBA has to guess and sometimes it probably can't find it. So try:

wdApp.CentimetersToPoints