Should I go into my boyfriends gmail or by what I have is enough already to know he’s not doing anything? ?

First of all, thats illegal. Second of all, that would be being a snoop. That shows that you don't trust him. That's very wrong. How would you like it if he did that to you? Show some respect and trust him. If he is loyal, he won't cheat. And if he does, you know what to do. But why would you do that anyway?!

you breaking law . means you cant be trustworthy for your boyfriend life. you seem like a very jealous girlfriend who cant trust anyone or any body  such your boyfriend. seem like you are very adgreed person who doesnt give a man any space for him self and cant have any friends and have any thing he wants. you dont have no rights to peek and check out his personality things that are private  and you dont have no right to spy on him when ever he is online or doing any thing without you being so nossy girlfriend. damn goodness he isnt married to you . i never do that to my husband . because its very wrong to do that to somebody. if you cant trust him with anything then he should break up with you find someone else better that can be trust . you just think too much about being cheat on. neither he isnt  doesnt means that you have the right to look and spy on him. if you are afraid that does happen  then its your faults that cant trust anyone else . if your boyfriend finds out that you had being doing something wrong and made mistakes then he wont give you second chance to be with. 

Obviously you know nothing about his LINUX or APPLE accounts.  Do what will BURN YOU.  You have my permission "witch"  Meanwhile he is in his bedroom banging your best friend.

That's not a very smart thing to do. You can risk facing serious jail time and fines for illegally going through his email account. If you sense your boyfriend was up to no good in the relationship, then the smart thing you can do is just walk away from him. Believe me, it's not worth the risk.

Why would you go into his gmail without his permission?

If you do not trust him and feel you need to you should split up now.  

How would you feel if he accessed your emails?

You should respect his privacy. I would fully understand why he would dump someone that doesn't trust him.

Your distrust is such that I suggest you end the relationship. If you must be this suspicious of a person, you're wasting both your time. Go elsewhere and leave him alone.

If you don't trust the person you are with, then you're 

with the wrong person. Regardless, everything you 

just said here makes it very clear that you are way 

too immature to even be in a relationship. 

 I see why men have depression

if you really do not believe you can trust him, then you should leave him. pretty simple.