There are two differences in your statements Let me / Let's - 'Let me' - you are asking the listeners permission to perform a task. 'Let's' (or Let us) - you are inviting the listener to participate in a task Check this email - you are just going to look at the email, maybe to see if is urgent. If you need further time following 'checking the email' it would be polite to ask the listener for more time. 'Check out this email' - You are going to check the email thoroughly. You are implying that you will read it thoroughly, or validate many facts within it. Also used in a light hearted manner 'Check out the size of this car!' or 'Check out the look on his face!" So your two statements differ quite a lot Let's Check out this email: "You and I will investigate the contents of this email thoroughly" or "you and I will enjoy the contents of this email together" if applied to something like a joke email. let me check this email: "excuse me for a moment while i just read this email to see if something requires my urgent attention"

You mean as a verb? Basically "check" means to look so as to make sure, and "check out" means to look to get your own opinion... to evaluate something. "Let's check out this email" - there's probably something of interest in the email and you want to know what it is. "Let's check this email" - you have probably read the email before, and remember that there may be some useful information in it. You want to look and make sure, or find that information. Between "let me check this email out" or "let me check out this email", both are correct and have the same meaning. Some phrasal verbs can be separated, and "check out" is one of them. However, if you use an object pronoun (me, them, it, her, him, etc) with a phrasal verb that can be separated, then the object pronoun can only go in the middle. "Let me check it out" is correct. "Let me check out it" is clumsy and wrong.

1 - No difference 2 - yes, you can say both 3 - you can't say "let me check out it" but you can say "let me check it out".

yeah I am sorry I made a typo and thanks a lot

I agree with Kebnov's answer. By the way, 'check' and 'check out' are verbs in all your examples.