[RPG] What happens to a Wizard’s Transmuter Stone when she dies

It's just a temporary inconvenience, except in a singular case.

Basically, all that a familiar's death means is that you'll have to recast Find Familiar at your earliest convenience (so it means the loss of a spell slot for a day if you have a time pressure and can't take the extra 10 minutes to cast it as a ritual). The spell/ritual does cost 10gp, but that's a pretty trivial amount for an adventurer. It also has a casting time of 1 hr so your party probably needs a short rest while you do it rather than simply doing it between encounters.

The only case where having a familiar come to harm is a permanent issue is in the rare event that you decided to retrain the spell (or as a wizard, have lost the spell book that contains it). These are the only times when losing a familiar has long term ramifications.

As to whether or not the newfound familiar obtained when you recast the spell is the same being reincarnated/resurrected, or a different being entirely, this is up to you and the flavor you choose for your familiar.

Alive or dead, it stays enlarged for the duration of the spell.

Unless otherwise stated, a spell lasts for its duration.

The only rule that covers general spell duration is a small section on page 203 of the PHB:

A spell's duration is the length of time the spell persists. A duration can be expressed in rounds, minutes, hours, or even years. Some spells specify that their effects last until the spells are dispelled or destroyed.

And the rules that cover monster death aren't very extensive either. Page 7 of the Monster Manual says, under "Hit Points":

A monster usually dies or is destroyed when it drops to 0 hit points. For more on hit points, see the Player's Handbook.

And PHB page 198 says, under "Monsters and Death":

Most DMs have a monster die the instant it drops to 0 hit points, rather than having it fall unconscious and make death saving throws. Mighty villains and special nonplayer characters are common exceptions; the DM might have them fall unconscious and follow the same rules as player characters.

Other than that, there's no specific rule that says spell effects end on a creature when it dies, unless the creature is the one casting and the spell requires concentration (PHB, p. 203), so they should carry on (exceptions might be made at the DM's discretion).

Thus, the creature would stay enlarged for the full 1-minute duration of the spell, whether dead or alive, unless your concentration is broken.