You would have to hire an attorney licensed to practice law in Pennsylvania to handle an estate being administered there. You may want to post this to attorneys located within the county where you need help to find another attorney to handle the matter.
I am very sorry for your loss. If the Will was not signed prior to death, it would not be a valid devise of property. It would seem that probate will be needed to handle your father's estate. You can try to contact the DMV and see if they will allow your mother to transfer your father's vehicle to herself using his death certificate as his heir and then she can transfer the vehicle. The vehicle should be exempt from any claims of creditors.
I am very sorry for your loss. The Estate may potentially need to go through Probate in the County where your mother passed, however that will depend on many different factors, one of which being how your mother and father held the assets. You should consult with an Attorney related to what process you will have to do related to your Mother's Estate.
You can file a creditors claim for back child support. Further if the children are minors you could have an attorney Petition for a family allowance which would allow you some funds to support any minor children. Also if minors they could be entitled to additional assets as well.
Sending the Waiver prior to the funds is common practice, however I would inquire about a informal accounting, the amount of your share and when it will be sent. You typically would recieve this information along with the Waiver. As a beneficiary you are entitled to request an actual formal accounting and not at all required to sign such Waiver.
You can not really guarantee an inheritance in the future. You could buy in to the company, the person giving the business shares could change their Will and not devise you the rest of the shares. You should be careful in proceeding and ensure you properly protect your rights to future shares in the business.
A Will does not need to be notarized to be valid. A Self Proving Affidavit attached to the Will would allow the Will to be admitted without finding the witnesses to the Will. However you can admit a Will without the Self Proving Affidavit, it just requires some additional steps. If your father does not have capacity at this time, you will not be able to do anything to correct or change the document.
The law in the state that you are serving as personal representative would apply related to expenses. Typically reasonable expenses will be covered and you can report those expenses to the beneficiaries.
In order to become Personal Representative you will have to Petition the Court to appoint you. If the Will listed you as a back up, you would have to open a Formal Administration with the Probate court and you then would have authority to transfer the bank account into an Estate Account. If there is no Will, you can petition the Court to be appointed and if no one contests the same, it would likely you would be appointed.
His children would inherit all his assets equally. Assuming he only has one adult daughter as you have indicated, she would inherit everything. However if he had any children who died before him, their children would be beneficiaries.