What's the difference between minimum feature size , transistor gate length and half pitch?
If you look at how actual IC fabrication is done you will see that the minimum feature size is the size or the width at which a transistor or any type of material on the silicon surface can be drawn at. This is usually analogous to the transistor gate length because the properties of the transistor depend on the ratio between the width and the length of the gate which essentially effects the flow of current through the junction. If the minimum feature size can be reduced, this means the transistor length can be reduced effectively making the transistor smaller with the same electrical properties. This allows for lower current flow between the junction for the same purpose and lesser heat dissipation.
Half-pitch is essentially the minimum distance between two features on the IC just like you mentioned.
The size of the transistor is best described by it's gate length. It's a little more complex than that as transistors normally require different ratios of width to length based on design requirements, so not all transistors on the IC will have the same width but they almost always have the same length which is analogous to the minimum feature size.