Why does old glass sometimes look purple?

Manganese was often used to clarify the glass mix. Over many decades, exposure to ultraviolet rays in sunlight causes the manganese to react. This slow chemical shift turns the clear glass a pale amethyst shade. It happens most frequently to pieces left sitting on bright windowsills.

I have a thick cut glass fruit bowl from my grandmother, but how can you tell if the cuts were actually done by hand or just pressed in a machine?

Run your thumb firmly over the patterned edges. Hand-cut facets feel sharp and precise because they were ground on a wheel. Pressed glass patterns usually feel slightly rounded or blunt. You will often find faint mould seams running through the design on a pressed piece.

Can I wash my antique glass bowl?

Warm water and mild soap are safe. Just keep the temperature moderate. Avoid sudden temperature changes. Plunging a cold piece into hot water can shock the material and cause it to crack.

We bought a blue glass bowl with white figures on it at an estate sale, and it feels a lot like porcelain, so what exactly is this type of glass called?

You are describing cameo glass. Makers applied a layer of opaque white glass over a dark blue body. A craftsman then carved away the white layer to leave those raised figures standing out against the background. It takes great skill to cut away the casing without breaching the base colour.