Short answer: No, there are currently no rules about what happens if you sleep in armour. The are rules about how fast you can don and doff armour [pg. 146 PHB] that's quite kind and fast (you do not don full plate in only ten minutes in real life) so characters shouldn't really have any issues with switching in and out of armour.

The are however rules for swimming in armour. Page 183 in PHB under the heading Suffocating. It's what you use to simulate drowning which is what happens if you try to swim in armour, especially the sort made primarily of metal.

Long answer: Sleeping in armour while not ideal or the perfect recipe for a great night's rest isn't as bad as you'd think. Anyone who ever played hockey or American football knows that laying down and rest in all that padding and equipment isn't bad. People all through history have actually slept and spend long times, sometimes months at the time, pretty much constantly wearing armour both day and night. The great siege at of Malta is one such example. Another example though a non-combat one, is how the moon astronauts slept in their space suits during breaks between the moon walks. Even now modern soldiers are sleeping while wearing body armour.

Now some might point out that modern battlefield body armour is designed to be comfortable. But that's just it. All armour through the ages has been designed to be comfortable to wear, to move in and to rest in because otherwise no one would be able to use it during long periods of combat. Full plate is both agile, flexible and feels light to wear. Second thing to think about is that humans are adaptable and can learn to endure the harshest conditions (can't speak for those poncey elves though). The human brain can actually tune out things it finds annoying or distracting over time. So even if it is uncomfortably to wear armour you forget it after awhile.

Which bring me to my point. (Finally right.)

If the reason you want rules for sleeping in armour is realism you don't really need rules as the effect of a night in armour would be negligible for a fit person trained in wearing armour. You might give the players a disadvantage for the first roll they make after waking up if they make sleeping in armour a habit. But the truth is that the cold, wet and fear of sleeping outside in a dangerous situation have much greater impact on how well rested you are than if you sleep in your armour or not.

However, while sleeping in your armour might not be that bad for your rest, prolonged armour use without breaks will eventually lead to all manners of unpleasant side effects. For example there is something called pressure ulcers also known as bedsores. They are incredible painful and nasty. Another nasty effect is spinal compression.

Now there might not be any rules for those effects either though if they were they would most like be in the form of sickness and injury than anything else. Still, Google some images of bed sores and show your players and they might start removing their armour a bit more often.

No

Every class proficient with martial weapons is also proficient with simple weapons, so there would be no mechanical advantage to using a trident over a spear (unless in some unforeseen circumstance you needed the extra 1lb).¹

There would be a reason to use a spear (simple weapon) and not a trident (martial weapon), but not the other way around. Bards, clerics, druids, monks, rogues, and warlocks are proficient with simple weapons but not martial weapons. For them, their proficiency bonus would be an advantage of the spear over the trident.

However, these are not the only weapons with the same mechanics (other than price and weight). The following martial weapons are mechanically identical:

• battleaxes and longswords are both versatile weapons causing 1d8 (1d10 two-handed) slashing damage
• morningstars and war picks both cause 1d8 piercing damage

1. The "Variant Entertainer: Gladiator" background (PHB, p. 131) allows a player to "replace the musical instrument in your equipment package with an inexpensive but unusual weapon, such as a trident or net." RAW this only gives the trident as equipment. However, a DM could house rule that it also grants proficiency (in place of the music instrument from the Entertainer background). If allowed and taken by a sorcerer or wizard, they would have proficiency with tridents but not simple weapons (i.e, spears).