The mouse doesn't show in Lemmix properly under Wine or CrossOver under normal conditions. Alternate solution could be to enable or disable hardware acceleration in winecfg, make sure you have the latest version of Lemmix as DullStar suggested, update to the latest version of Wine, or compile Wine from source.
Alternatively, use Lemmini, Tundra, or run Customized Lemmings in a DOSBox.