What happens if my mount's HP drops to 0 in combat?
Does it make death savings throws? Can I heal it and not have to spend gold on a new mount?
The rules for Death Saving throws are specific for Player Characters (PCs). Most DMs will not use them in other creatures.
From the Monster Manual, p. 7
A monster usually dies or is destroyed when it drops to 0 hit points.
Note that NPCs and usual beasts are described in the Monster Manual as well (Appendix A and B), so "monster" is kinda broad.
Other than that, there are DMs that use the Death Saving Throw rule for the Main Villain and other NPCs, allies or foes. I use them for anything that resembles a PC, as described in "Monsters and Death" at the end of the Death Saving Throws section:
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.
So, essentially, ask your DM whether he would treat your mount as a "special nonplayer character".
From the PHB p. 197
If damage reduces you to 0 hit points and fails to kill you, you fall unconscious.
While it's unconscious, you can heal it, like you can heal any PC.
You can't heal a dead creature. You can, however, resurrect it. Revivify, for example
Touch a creature that's died in the last minute. It returns to life with 1 hit point. This spell can't return life to a creature that died of old age, and it can't restore missing body parts.
Note that the cost for Revivify is 300gp, so buying a new horse is cheaper after all.
Usually it would die, but if there is sufficient (emotional) attachment the DM might give you a chance to save him. My DM gave Puppy, a youg worg I befriended after killing his brother and sisters, a death saving throw, because of the emotional connection that had built up between us. Also, I threatened with physical violence if Puppy died.... But thats probably unrelated :p So yeah, the DM's discretion is the main thing, as with most things, although consistency would be useful (so let everyone DS their mounts if they have RP-bonded)