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.