I'm greatly appreciate that you have agreed to help me
... this is grammatically wrong;
I am appreciating / I appreciate.... however, in this case "I appreciate" is better.
I greatly appreciate that you agreed to look after my house and pet while I am away.I will be leaving on the2nd. Juli, which means a week more from now.
2nd. Juli - 2nd July (July is the English term for Juli)
My staying in the south of the Country lasts for five days.
.... I think you need to use a more simpler tone for this letter;
I will be travelling in the South and returning home after five days.We can always Keep in touch by using my mobile number in case of emergency.
... give your number to him.