Yes, in fact there are several regional accents in Mexico, just as there are regional accents in the United States.
Mexico uses the acute accent.
México
Every country has it own accent. Denmark too.
Yes, Cuban Spanish has a unique accent, just as each Spanish-speaking country has its own distinct accent.
Mexico, but without the accent in the 'e'.
Latin America accent
El Zocalo, Ciudad De Mexico by the way there is an accent mark above the 'O' in Zocalo and an accent mark on top of the 'E' in Mexico.
Clasico (with an accent over the'a').
Yes
'Mexico' in Spanish is still 'Mexico.' It is pronounced differently, though. It is pronounced, "MAY-hee-co." It is written with an accent, "México", to designate the pronunciation is not "standard". Following normal rules, "Mexico", without a written accent, would be pronounced "meh-HEE-coh".
On the e: México.
When saying Mexico with a spanish accent, it often sounds like Meh-heek-oh.