Can "living constructs", such as the warforged from UA: Eberron, benefit from healing spells and the Medicine skill?
It's unclear whether they could be affected by healing magic, nor whether medical techniques could apply to a creature like a Warforged or Shardmind.
If they can, what would it look like?
In general, yes: a creature's type has no mechanical impact.
Cure spells work on creatures: constructs are creatures. However, all of the curative spells in the PHB (I think) specifically say they don't work on constructs or undead; this is not a general rule though.
Medicine works on "your companions": with a certain generosity of spirit, this extends to construct companions. Medicine allows you to stabilise or diagnose a disease. Since all constructs so far described are destroyed at 0hp, you can't stabilise those specific constructs. Nothing stops a construct contracting a disease, however, constructs are likely immune to organic disease; magical and construct only diseases are possible.
As to what it looks like: that's up to you.
So in general, yes, however, the specific cure spells and specific constructs so far published limit the general application.
Warforged and Shardminds are races very specific to the Eberron campaign setting.
There is no official 5e material as of this writing, but there is an Unearthed Arcana with a first draft of Warforged on the Wizards site. From that article:
Living Construct. Even though you were constructed, you are a living creature. You are immune to disease. You do not need to eat or breathe, but you can ingest food and drink if you wish.
It's not 100% clear if the line "living creature" means that you bypass the "construct" nature. That stated, the constructs in the MM have a large set of Immunities not given to the Warforged. So there's clearly a difference between them and "normal" constructs.
If you don't quite like the Unearthed Arcana version, there are several more "community" version here. Sadly, they use similar wording and offer no clarity.
As a DM, you're likely creating these races, so this decision is really up to you.
Treating a PC as full-on construct goes beyond just healing. So I would tread that lightly.
The mending cantrip specifically says it can repair constructs, but it doesn't give an HP value associated with that:
This spell can physically repair a magic item or construct, but the spell can’t restore magic to such an object.
The intro to the Monster Manual says, under "Type":
Constructs are made, not born. Some are programmed by their creators to follow a simple set of instructions, while others are imbued with sentience and capable of independent thought. Golems are the iconic constructs. Many creatures native to the outer plane of Mechanus, such as modrons, are constructs shaped from the raw material of the plane by the will of more powerful creatures.
The description of the warforged race in the Wayfinder's Guide to Eberron (p. 67) says, under the "Living Steel and Stone" heading (emphasis mine):
While they’re formed from stone and steel, warforged are living humanoids. Resting, healing magic, and the Medicine skill all provide the same benefits to warforged that they do to other humanoids.
Cure Wounds states, "This spell has no effect on undead or constructs", so it's out.
Medicine lets you stabilize dying companions or diagnose an illness. The illness part isn't going to help as a warforged's Living Construct trait makes them immune to disease. They can be dying though so the RAW says you help them out. RAI I think this makes sense as if a construct is able to be, "dying", somehow perhaps the general medical techniques (keep the fluids from coming out) would still apply.
You know, strictly speaking, neither the Celestial Warlock's "Healing Light" ability nor the Circle of Dreams Druid's "Balm of the Summer Court" ability explicitly exclude constructs or undead. Not sure if it's intended that way, but certainly useful if you've got Golems or Animated Dead in the party.
