There are two factors at play here. Line of sight and Clear path to target (what previous editions called Line of Effect).
Line of sight is only required for the first target. It's not specified as required for the secondary targets so you should not require it.
From "Targeting" on page 204 of the PHB:
To target something, you must have a clear path to it, so it can't be behind total cover.
If you place an area of effect at a point you can't see and an obstruction, such as a wall, is between you and that point, the point of origin comes into being on the near side of that obstruction.
So a valid target on the first part of spell is something you have a clear line of sight to, and also a clear path towards. The secondary targets do not need a clear path from you, but from the initial target since that's the point from which they emanate.
As far as your reading of the spell with respect to secondary targets, you're reading "As many as" incorrectly. This means "up to 3" not "exactly 3", so if you only have 2 available targets (that you want to hit), you're welcome to do so. While three bolts are created, they do not all need to be targeted and can be assumed to ground harmlessly.
The targets are chosen by the caster as it's not specified they are chosen randomly.
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.
Best Answer
Yes, this spell (and others worded the same way) can target worn and carried items
When a spell can't target worn and carried items, it says so in its description. For example enlarge/reduce which says:
Or catapult which says:
On the other hand, this spell has no such restriction. The only thing it says on the matter of targeting is:
Which, as you pointed out, has no further restrictions placed on it. Since there are no general rules prohibiting anything from targeting worn or carried items, the spell can simply do it.
This is likely Rules as Intended
Crawford has answered a general question about this in a Tweet:
DM can determine the effectiveness of this strategy
The obvious concern with this from a DM standpoint would be "is this fun and/or balanced?" and that has to be a decision made on a per table basis. However, it might be helpful to note that Crawford talked a bit about it as a continuation of the above tweet (if it matters at all to your table).
As a player discussing this with a DM, you might want to consider also that, while this would probably be a lot of fun for you, the player, to do against an enemy it might be less fun when the enemy is destroying your equipment. When a strategy is valid for one side, there is no reason the other can't also use it. Of course what is fun or not as I've mentioned before, will depend on the table. All I'm suggesting is that such a thing be part of the discussion.