How do you pronounce ě?
It's pronounced like the "ye" in "yesterday."
*Note: When "ě" follows the letter "m", it is pronounced as if there was a letter "n" between them. For example, the word "měkký" means "soft" and is pronounced "mněkký."
Well, not only that. In this very case "děkuji" is pronounced as [ď e k u j i], so the "d" is palatalized and the "e" is quite the ordinary [e].
See the Tips and notes or the Wikipedia article about Czech phonology.
When should I use Dekují and Diký?
Děkuji is more formal than Díky. Děkuju is in between and is very universal.
Why is the "a" pronounced velke a or is that an error with the app?
the latter, no need to report
Should "díky a dobry víkend" be ok?
No, no-one says dobrý víkend.
Melo by to byt dekuji a mejte hezky den
No, mějte hezký den is not used in Czech and it is an ugly anglicism.