1. From what I have learnt, it is the mass. You know that beta particles are electrons while alpha particles are helium nuclei. Consider a bowling ball and a tennis ball heading towards you and you wish to smack it away by hitting it perpendicular to do with a stick with the same strength. Which would be easier? Considering you will hit the objects with 100% accuracy.
The tennis ball will change its direction more easily rather than the bowling ball so we can say that electrons deflect more than alpha particles.
Or look at it from a physics perspective, using momentum : p=mv, (let us assume that the alpha particle and electron are travelling at the same speed for simplicity's sake), alpha particles will have more mass than electrons so it has more momentum, therefore it is harder to accelerate (change direction in this scenario) than the electron.
The charges are opposite so they deflect in opposite directions
Now the only thing here is that alpha has a higher magnitude of charge with 2(magnitude and the electron of 1 (magnitude). I haven't learnt this at school but I think it is because the difference between the magnitude of the charges is insignificant (1.6 x 10^-19) while the mass difference is relatively large (approximately 2 amu) therefore the mass difference overwhelms the charge difference and allows electrons to be deflected at a higher degree than proton
2. I would think that it would be impossible to distinguish between a particle accelerator and a cyclotron because they both require a charged particle. Perhaps another person can help you on this.