What is "stillness" or samatha? Is a "still" mind: a mind without any thoughts; or a mind without a certain style of thoughts; or a mind which has lost a certain way of connecting to thoughts?

I assumed it was a lessening of conscious habit rather than thought per se.
All thought is movement and a still mind is one that has no movement in it except when required.

When you need to use the memory - "what's 1+1" or "where's my car keys" - the mind provides the answer. It's only the innecessary thinking that stops. Innecessary thinking is thinking that isn't required by the immediate situation.

This also means thoughts don't have the same energy anymore - the energy to make you feel emotional for example.

99.9% of thoughts in a normal undisciplined mind are created by habit, circumstances and old emotional pains.