I would like to know wether most chinese dialects agree on a similar pronunciation/name of the country (中国/Zhōngguó) as in Mandarin, or there are some dialects who call it with a totally different (unrelated to mandarin) name.
Do most Chinese dialects agree on a similar pronunciation of 中国/Zhōngguó as in Mandarin?
Dialects do not agree as such. They pronounce 中国 the same way they pronounce the two characters 中 and 国 separately.
But if you mean, Do most Chinese dialects have a similar pronunciation of 中国/Zhōngguó as in Mandarin?
Then, it depends on what you mean by "similar". Since the various Chinese dialects/languages belong to the same language family, they do sound similar to an extent. There are also quite obvious differences. As an example, southern dialects such as Cantonese and Fujianese/Taiwanese have kept the clipped tone (syllables ending in p/t/k), whereas northern dialects have mostly lost it over time. In Cantonese, the word 国 ends in /k/, not a vowel as in Mandarin. It does sound quite different from Mandarin. So it depends on your definition of "similar".
Are there some dialects that call 中国 with a totally different (unrelated to mandarin) name?
I do not think so, but I do not claim to know all of the dialects/languages under the huge umbrella of "Chinese language". If there are people who are aware of such dialects, feel free to correct me.
China, (meaning the "People's Republic of China", or "Mainland China") is, in its written form at the present moment, "中國", regardless of your Dialect, Ethnic origin, or political inclination.
However, 中國, is pronounced differently by different Dialect, Ethnic Groups as follows:-
Cantonese ---- Zung1 Gwok3,
Hokkien / Fujian / Min Nan ---- Tiong Kok
Hakka ---- Chûng-Koet
TeoChew / Chaochow ---- Dong Gog
These are just some of the Dialectical / Ethnic Groups. There are more of course. I am no expert, just what I know.