When I look at a monster it has several attributes listed with it. I understand the "early starter", "late starter" ones, but there are others that don't really seem to be defined anywhere. Things like "Meleeproof" or "Noisy". What do these attributes mean, and is there a way to view the definitions in game? Does a monster's type determine its attributes, or can they change?
The three most important imo are
The rest all make sense based on what they are called. Examples: Hearty means the monster will have a higher HP than a monster without it. Strong means a monster will have higher strength. Pinch Hitter means the monster can learn abilities that can be activated when in danger, etc.
There are some that are element+prone which means the monster is weaker against that element. Some are element+proof which means the monster has a strong resistance to that element or will learn resistance abilities for that element.
For completeness: As mentioned by Yuck, only the first three attributes define the monster. The fourth attribute has no in game effect for your monster and is there for fun. See Yuck's answer for more information.
In addition to Adanion's answer the fourth attribute listed is just for fun. It has no in-game effect and is just meant to be cute or entertaining. Some of these are actually Cute, Zippy, and so on.
Only the first three attributes define the monster.