[RPG] Regeneration while dying in 4th edition

First of all, this is not so much a problem, as a design decision. 4e is purposefully designed to let characters start afresh every morning, to make encounter design easier for the DM (and published adventures). Another thing worth noting is the fact that hit points are highly abstract, and don't necessarily represent physical damage characters sustain:

Hit points represent more than physical endurance. They represent your character’s skill, luck, and resolve—all the factors that combine to help you stay alive in a combat situation. - PHB293

In fact, as the term bloodied suggests, until the character has lost half of their hit points, they're not even that. With this out of the way...

Two ideas come to mind. One is to limit the amount of healing surges a character regains with each extended rest. Probably a flat value modified by sleeping conditions and available medical care, something along the lines of:

Everyone regains 3 healing surges after an extended rest. Following circumstances may increase this number:

+1 - Moderate Heal check / +2 - Hard Heal check

+1 - Moderate Endurance check / +2 - Hard Endurance check

+1 - Decent sleeping conditions (warm bed roll, rations & water) / +2 - Excellent sleeping conditions (hard bed, hot meal and grog)

These numbers, of course, are not at all tested, and can be modified according to your preferences. This doesn't really change the balance of the game, as long as you're careful with the number of encounters you throw at the tired party.

The second idea is to award a wound condition to a character every time they drop below 0 hp. Broken leg, -2 to speed. Fractured arm, -1 to attack rolls, etc. Each damage type could impose its own wound - WFRP had something along these lines, IIRC, and so can be used for inspiration. Healing these wounds may require time, ritual magic, or burning healing surges at the end of an extended rest (in which case these two can be combined). Note that this actually changes the way characters operate, and so should be approached with extreme caution.

No.

The fifth edition of D&D is effectively an entirely different game from Fourth Edition. While there are some similarities (roll a d20 and add stuff), they tend to be pretty superficial.

Even when names are reused, the concepts underlying the name can be entirely different. For example, a saving throw in Fourth Edition is a difficulty ten roll without modifiers at the end of your turn. In Fifth Edition, it is an ability check usually made at the time something is first affecting you.

With that being said, there is a great deal of overlap in terms of the world (sort of, it's complicated), flavor, and "soft" advice between the two systems. You could certainly use the situations, locations, and characters from Fourth Edition to inspire the same in a Fifth Edition game... You'd just need to throw out all of the Fourth Edition mechanics and rebuild them in 5th edition terms.