How should I clean these old glasses?

Wash them by hand in warm soapy water. A soft sponge clears away most residue without risking the surface. Never put them in a modern dishwasher. The high heat and harsh chemicals will etch the glass over time, leaving a cloudy finish you cannot remove.

I have heard that a wide bowl makes the wine go flat faster than a flute, so why did people ever use them for sparkling drinks?

Earlier sparkling wines were much sweeter than modern varieties. The wide bowl allowed the carbonation to dissipate quickly, which softened the heavy yeast and sugar notes on the palate. Drinking habits were also different. People drank smaller servings at a faster pace during toasts, meaning the liquid rarely sat long enough to go flat.

Can I use these for cocktails?

Yes. The shape works well for drinks served up without ice. The wide rim allows the alcohol fumes to disperse, making them a staple in many modern bars.

When I look at older sets of stemware, the heights and rim thicknesses seem to vary quite a bit from glass to glass. Is this normal?

Minor variations in height and thickness are standard for hand-blown glass. Craftsmen relied on their eyes and hands rather than automated machinery. The amount of glass gathered on the pipe dictated the final thickness. Two gathers of slightly different sizes meant the resulting bowls would not match. Even pieces from high-end factories show slight discrepancies. This variation sets manual production apart from modern factory-moulded uniformity.