The answer is yes, Animate object would work on a corpse. The exact effect would depend on the size of the corpse.
While there are specific defined terms in D&D 5e there are also a equal number of that rely on what the word means in English.
Object - a material thing that can be seen and touched.
Creature - an animal or person.
However there is a caveat. In various effects, powers, and abilities. The D&D 5e rules are consistent in referring to creatures as things that are living or animate. Objects as inanimate things like tables, chairs, rocks, books, feathers, etc. It not spelled out but it is consistent.
The things to remember is that D&D 5e rules are not to function as a wargame. They do not define the boundaries of what is possible during a campaign. The setting is what defines that. Instead they are a tool to aid the referee in adjudicating the action. For example the description of humans don't spell out every detail that could come up. The mechanics about humans are those that the authors feel that are useful or come up often. The important of which is the effect being human on character creation. The author expect referee to use what they know about humans to adjudicate anything that the rules don't cover because it is implied that humans in a D&D setting are just like people in real life only living in that world.
One implication of this is that animate objects doesn't change any other physical property of the object other than to animate with the stats provided. If you were to say animate a block of salt, possible considering what salt miners carved out of their mines, and it was to walk into water, then it is reasonable to rule that it would be affected adversely as salt dissolves in water. Perhaps by treating water as a acid attack on the animated object.
So a corpse animated as a object would still be a corpse and subject to decay, smelling bad, etc. It would not gain the benefits of being undead although at first glance it would be hard pressed for a character to tell the difference. One area where I can see the difference being important is trying to animate a skeleton. It is reasonable to assume that the various create undead spells joins the bones together to form a complete animated skeleton. While a long dead skeleton is merely a pile of separate objects of bone.
For stuff that has no real world analogue, elves, magic, etc. The authors expect the referee to fall back on their knowledge of the fantasy genre. Because the implied assumption that D&D is being used to depict a fantasy setting. Which is why they included a list of inspirational works in Appendix E on page 312.
In fantasy it is tradition for some spells to work on anything, a lightning bolt doesn't care if its target is a person, animal, or a piece of furniture. Some spells to only work on people, for example charming or enchanting a princess. And other spells to work only on objects, like the animated furniture from Fantasia.
First, this is a concentration spell, if your concentration breaks (p.203 PHB) the spell ends. It is also time limited - you can concentrate for up to 1 hour (1st or 2nd level slot), 8 hours (3rd or 4th level slot) or 24 hours (5th+ level slot).
Second, it does exactly what it says:
If the target drops to 0 hit
points before this spell ends, you can use a bonus action
on a subsequent turn of yours to mark a new creature.
So if a creature drops to 0 hit points and you keep concentrating then on a subsequent turn of yours (which can be in a completely different encounter 45 minutes later) you can use your bonus action to mark a new creature. And so on and so forth until the end of the spell.
It must be cast on a creature and can stay active beyond that encounter
So long as you can keep concentration, meet the original requirements of casting the spell, and the previously cursed creature has been dropped to 0 HP, you can use a bonus action to curse a new creature. No requirement to keep it active (just concentration) or for it to be the same encounter.