1) The OED has ‘cool-trough’, first recorded in 1777 in a context where the meaning is clearly the same as that of the ‘coultrough’ used in the Sheffield smithies.
The prefix is taken by the editors at face value and the word is explained as a ‘trough in which anything is cooled’. The Yorkshire evidence is much earlier: in his will of 1510, transcribed in the Ecclesfield parish register, Thomas Parker of Whitley left his water wheel at Whitley and two stone troughes called Coltroughes to his son John: in 1542 Richard Boyer of Sheffield bequeathed to his son and John Hobson all his smythe gere and the coltroughe. There are many additional references in the inventories of cutlers from the seventeenth century: in 1689 Francis Brownell, scissorsmith, had a Stithie and Stock and 2 Coultroughs and in 1709 Joseph Webster had A Stiddy and Stock Cow trough and glaszer. Some of these spellings raise a doubt about the usual interpretation and the explanation of the prefix may need to be revised. One possible connection is ‘cowl’ in the sense of a tub or vessel for water, in which case the source is likely to be the Middle English cuvele.