Do cell phones and email exist in Gankutsuou?
Gankutsuou's is an adaptation of The Count of Monte Cristo written by Alexandre Dumas and published in 1844. This is shown by the series' full English title: Gankutsuou: The Count of Monte Cristo.

The events of the original book take place in late 1830s France and as such the characters did not have access to technologies such as the telephone which would not be invented until the 1870s.

The futuristic setting of Gankutsuou is an adaptational choice and is very different setting to the original novel. This massive change in setting creates Adaptation-Induced Plot Holes such as this.

So your final question would be correct: Advancements such as cell phones are simply ignored in order for Gankutsuou to properly follow the plot of the original novel.