Can I install rubber speed cushions on my street?

NO, it is illegal for private citizens to install hardware of any kind on a public road. Traffic calming devices which includes speed humps, speed cushions and chicanes etc. are the responsibility of the relevant roads authority or local council in your area. Best to start with making an enquiry with your local council or speak to your local member. It would also be wise to have the support of your neighbours as well, and maybe even that local primary school or childcare centre around the corner to add weight to your request.