Note: The answer to the question is not the same in every U.S. state and territory. I have provided definitive answers only when there is unanimity or near unanimity.
Did the deceased's family obligate themselves by not returning the the
first service, that of the Notice of Appeal?
No.
Unless they expressly agree to do so in writing to be responsible for a decedent's debts (which almost never happens in the U.S.) next of kin of have liability regarding the debts of a decedent (at least by virtue of being next of kin, obviously, if they were, example co-defendant in a case who participated in the wrongful conduct, that could be another matter), except that it may result in a claim against the probate estate which reduces the size of their inheritances (potentially to zero), if timely asserted in the probate estate.
As the next of kin are they obligated to receive the mail addressed to
the deceased?
No.
Also, counsel for a decedent are terminated as counsel as a matter of law upon the death of a client.
It is customary for a lawyer in the case (often, although not necessarily, the lawyer for the decedent if the decedent was represented by counsel at the time of death) to file a document usually entitled "suggestion of death" in the pending case when someone dies.
But, no one is obligated to do so, and if the party is self-represented in the case (a.k.a. pro se), or that party's lawyers have withdrawn from the case, neither the decedent's former lawyers, nor anyone else in the case, may even be aware of that fact that the party to the case has died.
What would be a possible remedy to keeping the claim active against
this defendant albeit deceased defendant?
File a claim in the estate of the decedent if the deadline for filing claims has not lapsed (which two years later, it may very well have).
Normally, either a claims deadline that functions by operation of law (e.g. a state law non-claim deadline one year after the date of death), or a claim deadline arising from publication of a notice to creditors in the legal section of a newspaper in the area where the decedent died in a manner prescribed by state statute, would bar the claims, especially if the status quo was that they had been dismissed at the time of the decedent's death, even if there were still undistributed assets left in the probate estate after the deadline for filing claims has expired.
It may also be possible to do a substitution of parties of the decedent's probate estate for the decedent in the pending case, if a probate estate has been opened and the deadline for filing claims has not yet lapsed.
If a probate estate has not been opened, usually, after a certain amount of time, a creditor may open up the estate without the consent of the next of kin, or a public administrator will be appointed if there are no next of kin who have done so, and there are assets left to be managed in the probate estate.
The law governing exactly when a probate estate has liability for the debts of a decedent incurred during life is a matter of state law that varies in significant detail between different states within the United States and is quite technical. I've written a couple of Colorado Bar Association journal articles on that subject applying Colorado law (Andrew Oh-Willeke, "Creditor's Rights In Probate - Part I and II", The Colorado Lawyer, May 2015 and June 2015).
Note that the general rules may not apply in all cases to lawsuit brought "in rem" (i.e. adjudicating rights in a particular piece of property) although those kinds of cases aren't very common in federal court and would rarely involve the fact pattern set forth in the original post.
Footnote Re Federal Civil Procedure
For many years there was a lack of clarity, and/or a split of authority between U.S. Court of Appeals Circuits, over the proper time to appeal a ruling dismissing a party entirely from a case as a matter of federal civil procedure in various circumstances. It was resolved a few years ago, but I don't recall the outcome of that case.
The general rule, expressed in Federal Rule of Civil Procedure 54(b) is that the dismissal of a party in a case cannot be appealed unless the trial court certifies that decision as final for purposes of appeal which is a discretionary decision for the trial court judge.
