It will not include the day it was sent, service is deemed to take effect within certain timescales, depending on how you served it. Also a warning delivered after 4pm on a working day is deemed to have been served the next day. One delivered after 12pm on a Saturday is deemed to have been received the following Monday.
The solicitors will probably allow your fees to be taken from the estate at the end of the case. However, depending on what type of case it is, they may ask you for funds on account.
No you cannot refer it to the Will Writing company.
This is complicated you must take professional legal advice.