No. The text of Verpine Shatter Rifle States that it replaces turn. Therefore you cannot use any other ability, attack or movement credited to the character.

Correct. You can't use Double, but you can use Twin.

A turn is normally broken up into two parts: move and attack. If you don't attack, you can move twice as far (double your speed). If you don't move and you have Double Attack then you can attack twice.

However, with a 'replaces turn' ability, you lose both your opportunity to move and to attack. VSR replaces turn. It just so happens that VSR itself generates an attack a part of its ability, but Double Attack doesn't interact with VSR - Double Attack is triggered only when you're making a normal attack without moving.

Twin Attack is different - it triggers after you make an attack, every time. It is not related to the move/attack parts of your turn. So when VSR generates an attack, Twin triggers as well. Twin also triggers on attacks of opportunity. The only attacks that don't trigger Twin Attack are those attacks generated by Twin Attack. (Otherwise, you'd have an infinite series of attacks... Attack, Twin, Twin off the Twin, Twin off the Twin off the Twin, etc.)