A few thoughts:

It sounds like his teacher should help him to develop an organization system so that he is capable of being responsible for his work. As for the lying, it is possible that he is not sure what else to do- you stated that he is sensitive and emotional and hard on himself so phrasing a question so that he has to answer in a way that implicates himself for getting in trouble is very stressful. Can you talk to the teacher about receiving more consistent communication? Incidentally, I hate those card systems, they are punitive and there is a large body of research to support positive behavior supports.

I would recommend an eval, probably both a psycho-educational eval and an occupational therapy eval. They should be able to do both at school. It sounds like he is having lots of trouble in noisy environments which could be an indicator of sensory processing issues or attention issues. Actually, reading your next paragraph, I would ask for a speech/ language eval as well- the following directions, answering questions are receptive language skills. A good speech pathologist will also report the interrupting; a child his age should have the social language skills to refrain from that much of the time.

He may actually need the video game time to recharge himself. I'm not big on lots of screen time, but some kids really need it and benefit from it. Wearing my professional (rather than my parental) hat, I would not try to 'catch him lying'. Rather, i would try and help him learn to be honest. make it a learning experience. If you know he is being dishonest about something that happened in school and you know what actually happened through teacher communication, you can tell him, "No, you should tell me what actually happened. You can say [insert event here]".

This is an excellent reference for positive behavior support. The model was developed at Vanderbilt University.