I am sorry about your loss.
Under Massachusetts law, a will is required to be published in the probate court in which the decedent was "domiciled" (as of his or her death) within 30 days of the date of death. Shortly after publication of the will, all heirs at law (usually spouses and children) are required to be sent a notice of both the probate proceeding and the requested appointment of the executor nominated by the will. That notice provides a date by which you or your attorney must file an "appearance."
This date is critical; if you do not file a notice of appearance by that date, you lose the ability to contest either the nomination of executor or the validity of the will. The notice of appearance is nothing more than a document with the name of the court, the case number and a title "Notice of Appearance" and stating that you are appearing in the matter.
There is then a second and even more critical deadline. That deadline is the date by which you must file what is known as an "affidavit of objections." The affidavit must be filed no later than 30 days after you file the notice of appearance. In the affidavit of objections, you are required to state under oath all of the reasons why you believe the will is invalid and/or the executor should not be appointed. These could range from lack of mental capacity to undue influence or even forgery to, in the case of executor appointment, a prior conviction of a crime involving fraud or theft.
I highly recommend that you consult with an attorney at least for the preparation of this affidavit (which must contain certain "magical" language), if not for the filing of the appearance.
I am sorry for your loss, and would answer your questions as follows:
1) Anyone has the right to see a will once it has been placed on file at the Probate and Family Court, unless it has been impounded by the Court for some reason. (This would be very unusual.) This should be the Court located in the county in which your mother was living at her death. The executor will often provide a copy to the heirs-at-law as a convenience, but it is not required.
2) Nobody needs a lawyer to see a will.
3) Yes. Simply go to the Court at which the will was filed, find the Docket (file) number at one of the computer terminals, and ask at the desk to see the file. The death certificate will have been filed with the will, and will be in the file.
4) The Court provides formal oversight, but not typically on its own. The executor is required to file with the Court an inventory of the state, as well as accounts showing all the property coming into the estate, any income, losses, sales, etc., and finally, all disbursements, to whom, and for what reason. The inventory and accounts must be filed within certain time limits, but there is no penalty for failing to do so. If a filing deadline is missed, the Court will neither know nor care. A petition to require the filing of an account or inventory can be brought by an interested person (someone taking property under a will, or a creditor), and the executor will then be required to file, or risk being held in contempt of Court.
5) To contest a will, see Attorney Fanger’s excellent explanation of the process.