The rules for Cover, Dodge and trivial shields should be enough
Before introducing new mechanics and homebrew new items, answer these three questions:
- What problem do you solve by introducing this new rule?
- Can this problem be solved by using existing rules?
- What possible balance issues this new rule might cause?
If there is no specific problem to solve, and you just don't know how to resolve a thing that your player has just announced, PBH page 6 "How to play" proposes the "fiction first" approach:
- The DM describes the environment.
- The players describe what they want to do.
- The DM narrates the results of the adventurers' actions.
When your player says "I use X as an improvised shield", and you don't know how to resolve this, because there are no rules for improvised shields in 5e, apply this "fiction first" approach. What this player are trying to achieve and how? Then use the common sense and existing rules to adjudicate the situation.
froze some water into a roughly shield-like shape, what would be the rules for using such an item as a shield?
Use Cover mechanics when the player announces "I freeze some water and hide behind it", this improvised shield is called "cover". Hiding behind something big enough gives you bonus your AC and saving throws:
Walls, trees, creatures, and other obstacles can provide cover during combat, making a target more difficult to harm. A target can benefit from cover only when an attack or other effect originates on the opposite side of the cover.
A target with half cover has a +2 bonus to AC and Dexterity saving throws. A target has half cover if an obstacle blocks at least half of its body.
They even can carry this obstacle with them, providing it is light enough, but picking it up and setting in the same turn should probably expend their action.
broke off a wooden door
Use Dodge mechanics when the player announces "I'm desperately trying to avoid being hit, using any suitable thing around as a shield". This sounds as the perfect opportunity for the Dodge action. Resolve this using the Dodge mechanics:
When you take the Dodge action, you focus entirely on avoiding attacks. Until the start of your next turn, any attack roll made against you has disadvantage if you can see the attacker, and you make Dexterity saving throws with advantage.
Last thing — if your players want to use something in an unusual way, and you are not sure how to homebrew this, just "reskin" an existing item. At least, this does not break anything.
hold up a metal lid
Use shield mechanics when a player found a perfect shield equivalent and wants to use it as an improvised shield. Just use rules for a normal shield in this case: +2 AC, needs an action do don, occupies a hand.
Would it be the standard +2 from a regular shield? If so, what's stopping someone from just using that as a shield from now on instead of buying on from a shop?
Nothing. Is it a bad thing? Why? Characters will find gold in their adventures anyways, and a shield costs only 10gp. Giving a "free" shield isn't worse than giving free 10gp, and normally you buy a shield just once. A knight using utensil in an unusual way could be more interesting than one with a "proper" shop-bought armor.