SQL Server Pivot Table with multiple column aggregates

I would do this slightly different by applying both the UNPIVOT and the PIVOT functions to get the final result. The unpivot takes the values from both the totalcount and totalamount columns and places them into one column with multiple rows. You can then pivot on those results.:

select chardate,
  Australia_totalcount as [Australia # of Transactions], 
  Australia_totalamount as [Australia Total $ Amount],
  Austria_totalcount as [Austria # of Transactions], 
  Austria_totalamount as [Austria Total $ Amount]
from
(
  select 
    numericmonth, 
    chardate,
    country +'_'+col col, 
    value
  from
  (
    select numericmonth, 
      country, 
      chardate,
      cast(totalcount as numeric(10, 2)) totalcount,
      cast(totalamount as numeric(10, 2)) totalamount
    from mytransactions
  ) src
  unpivot
  (
    value
    for col in (totalcount, totalamount)
  ) unpiv
) s
pivot
(
  sum(value)
  for col in (Australia_totalcount, Australia_totalamount,
              Austria_totalcount, Austria_totalamount)
) piv
order by numericmonth

See SQL Fiddle with Demo.

If you have an unknown number of country names, then you can use dynamic SQL:

DECLARE @cols AS NVARCHAR(MAX),
    @colsName AS NVARCHAR(MAX),
    @query  AS NVARCHAR(MAX)

select @cols = STUFF((SELECT distinct ',' + QUOTENAME(country +'_'+c.col) 
                      from mytransactions
                      cross apply 
                      (
                        select 'TotalCount' col
                        union all
                        select 'TotalAmount'
                      ) c
            FOR XML PATH(''), TYPE
            ).value('.', 'NVARCHAR(MAX)') 
        ,1,1,'')

select @colsName 
    = STUFF((SELECT distinct ', ' + QUOTENAME(country +'_'+c.col) 
               +' as ['
               + country + case when c.col="TotalCount" then ' # of Transactions]' else 'Total $ Amount]' end
             from mytransactions
             cross apply 
             (
                select 'TotalCount' col
                union all
                select 'TotalAmount'
             ) c
            FOR XML PATH(''), TYPE
            ).value('.', 'NVARCHAR(MAX)') 
        ,1,1,'')

set @query 
  = 'SELECT chardate, ' + @colsName + ' 
     from 
     (
      select 
        numericmonth, 
        chardate,
        country +''_''+col col, 
        value
      from
      (
        select numericmonth, 
          country, 
          chardate,
          cast(totalcount as numeric(10, 2)) totalcount,
          cast(totalamount as numeric(10, 2)) totalamount
        from mytransactions
      ) src
      unpivot
      (
        value
        for col in (totalcount, totalamount)
      ) unpiv
     ) s
     pivot 
     (
       sum(value)
       for col in (' + @cols + ')
     ) p 
     order by numericmonth'

execute(@query)

See SQL Fiddle with Demo

Both give the result:

|             CHARDATE | AUSTRALIA # OF TRANSACTIONS | AUSTRALIA TOTAL $ AMOUNT | AUSTRIA # OF TRANSACTIONS | AUSTRIA TOTAL $ AMOUNT |
--------------------------------------------------------------------------------------------------------------------------------------
| Jul-12               |                          36 |                   699.96 |                        11 |                 257.82 |
| Aug-12               |                          44 |                  1368.71 |                         5 |                 126.55 |
| Sep-12               |                          52 |                  1161.33 |                         7 |                  92.11 |
| Oct-12               |                          50 |                  1099.84 |                        12 |                 103.56 |
| Nov-12               |                          38 |                  1078.94 |                        21 |                 377.68 |
| Dec-12               |                          63 |                  1668.23 |                         3 |                  14.35 |