1) The error message is clear: number of active TXs at the end of SP should be the same as number of active TXs at the beginning.
So, when at execution of dbo.spLogger begins the number of active TXs (@@TRANCOUNT) is 1 if we execute within this SP the ROLLBACK statement this'll cancel ALL active TXs and @@TRANCOUNT becomes 0 -> error/exception
2) If you want just to avoid writing IF @@TRANCOUNT ... ROLLBACK within every CATCH block of every user SP then don't it. I would call dbo.spLogger within CATCH block after ROLLBACK.
3) If I have to call SPs from other SP using TXs then I would use following template (source: Rusanu's blog)
create procedure [usp_my_procedure_name]
as
begin
    set nocount on;
    declare @trancount int;
    set @trancount = @@trancount;
    begin try
        if @trancount = 0
            begin transaction
        else
            save transaction usp_my_procedure_name;
        -- Do the actual work here
lbexit:
        if @trancount = 0   
            commit;
    end try
    begin catch
        declare @error int, @message varchar(4000), @xstate int;
        select @error = ERROR_NUMBER()
                                   , @message = ERROR_MESSAGE()
                                   , @xstate = XACT_STATE();
        if @xstate = -1
            rollback;
        if @xstate = 1 and @trancount = 0
            rollback
        if @xstate = 1 and @trancount > 0
            rollback transaction usp_my_procedure_name;
        throw;
    end catch   
end
with few small changes:
a) SET XACT_ABORT ON
b) I would call dbo.spLogger within CATCH block only when there is @@TRANCOUNT = 0:
IF @@TRANCOUNT = 0
BEGIN
    EXEC dbo.spLogger ... params ...
END
THROW -- or RAISERROR(@message, 16, @xstate)
Why ? Because if dbo.spLogger SP will insert rows into a dbo.DbException table when one TX is active then in case of ROLLBACK SQL Server will have to ROLLBACL also these rows.
Example: 
SP1 -call-> SP2 -call-> SP3
                         |err/ex -> CATCH & RAISERROR (no full ROLLBACK)
              <-----------
              |err/ex -> CATCH & RAISERROR (no full ROLLBACK)
 <-------------
|err/ex -> CATCH & FULL ROLLBACK & spLogger      
4) Update
CREATE PROC TestTx
AS
BEGIN
    BEGIN TRAN -- B
    ROLLBACK -- C
END 
-- D
GO
-- Test
BEGIN TRAN -- A - @@TRANCOUNT = 1
EXEC dbo.TestTx 
    /*
    Number of active TXs (@@TRANCOUNT) at the begining of SP is 1
    B - @@TRANCOUNT = 2
    C - @@TRANCOUNT = 0
    D - Execution of SP ends. SQL Server checks & generate an err/ex
        Transaction count after EXECUTE indicates a mismatching number of BEGIN and COMMIT statements. Previous count = 1, current count = 0.
    */
COMMIT -- E - Because @@TRANCOUNT is 0 this statement generates 
another err/ex The COMMIT TRANSACTION request has no corresponding BEGIN TRANSACTION.
    -- End of Test
5) See autonomous transactions: it requires SQL2008+.
An Autonomous transaction is essentially a nested transaction where
  the inner transaction is not affected by the state of the outer
  transaction. In other words, you can leave the context of current
  transaction (outer transaction) and call another transaction
  (autonomous transaction). Once you finish work in the autonomous
  transaction, you can come back to continue on within current
  transaction. What is done in the autonomous transaction is truly DONE
  and won’t be changed no matter what happens to the outer transaction.