So please check the below and find out what's wrong (haha.. maybe something I am not aware). Actually I occasionally made some grammar mistake; so I am not here to criticize someone's English writing. :)
OK, here is my answer. Please correct further if there is anything inappropriate.
1. Peter doesn't like Tempura. I don't like it also (should be "either" instead of "also") If the sentence is in a negative tune, you should say "either".
2. I would suggest you to try Pad Thai in this restaurant. (... suggest you "try". No need to put "to" before try. Same as let - don't say "let you to try". This is quite Chinese English. Or in a more proper form. You can say "suggest you should try".
3. Paul can not finish his homework by himself. (should be "cannot" one word.)
4. Don't shut the window or close the door. (Same as the above point 1, if the sentence is in a negative tune, it should be ".... window nor close the door.". If you say "or" that means it is in a reverse tune. So the meaning would be "you don't need to close the door". However, the use of "or" depends on the verb. If a sentence like "You can't drink coffee or tea because it upsets your stomach". Mmmh... now I recalled my computer learning on the Apple II computer (year 1983), in the computer language, even "and" should be used as "nand" for negative tune. :)
5. Mary is looking forward to see you in the coming weekend. (.... forward to seeing.. instead of "see". In some cases "to" is a preposition; so after "to" should be a gerund or noun.
6. I resent that email to you last Friday. Please check it again. ("resent" is another verb actually - means "get angry of" or "feel anger with". Don't often put "re", "dis", "un", "non" before a verb if you are not sure about the proper usage. Or if you are in a rush, you can try to write "re-sent". Actually this is quite Chinese-English. I would suggest writing it as "I sent that email to you again last Friday. Please check it.".