The repair light/moderate/heavy/critical damage spells from the Eberron books?
Will work as cure light/moderate/heavy/critical wounds.
Sor/wiz, artificer and spellthief only. A cleric can get access to some of these by choosing the Warforged domain.
Also, Total Repair (artificer 6 and warforged domain 8) is the equivalent of heal, and should keep you going for a long time.
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.
Best Answer
As of 5e's published release and the Unearthed Arcana article, Warforged are treated no differently than PCs of other races.
Since it does not mention any healing exceptions for warforged in the Unearthed Arcana article that introduces them to 5e; they should have any and all healing effects work on them as per usual.
2018 update (thanks @Trevel)
This question has now been explicitly answered in the Wayfinder's Guide to Eberron, under the description of the warforged's composition (p. 67):