How to expressing grammar "may" and "can" in Korean?

Question

Answer

I can do this.
나는 이것을 "할 수 있습니다."

You may do this.
너는 이것을 "해도 됩니다."

You can/may smoke here.
당신은 여기서 담배를 피울 수 있습니다.
당신은 여기서 담배를 피워도 됩니다.

Answer

You should listen to TTMIK lessons. Can has at least two meanings in English.

ㄹ/을 수 있다 and -ㄹ/을 수 없다 - used when expressing or asking for capability. Can you cook food?
http://www.talktomeinkorean.com/lessons/level-2-lesson-17/

May/can -도 되다 - permission. Can i eat this?
http://www.talktomeinkorean.com/lessons/level-4-lesson-8/

as for your other questions.. listen to all the lessons. they have a lesson on should/shouldn't.