The word Mexico does have an accent in Spanish, but not in English. There are no accents in English; the words in which we are used to seeing them (resume, fiancee) are actually words from other languages - usually French - that have been loaned into English.
The word "Mexico" looks so similar in Spanish and English that I don't think it matters if you use the accent when writing in English.
On the e: México.
Mexico uses the acute accent.
México
The accent is on the first syllable in the word "compound."
If you are talking about an accent mark called the 'accute accent', then no. It does not.
an accent is 'un accent' in French
No, the word "comprare" does not have an accent mark.
No, there is no accent in the Spanish word "capítulo."
Mexico, but without the accent in the 'e'.
The English word "arithmetic" carries no accent mark. The equivalent Spanish word 'aritmetica' has an accent over the 'e'.
Latin America accent
I just love your accent!