I'll try and answer each of your points in turn.
I'm assuming that you are referring to fortress mode.
Is it better to wear multiple pieces on one bodypart?
It is better to get your dwarves wear multiple pieces of clothing on one body part to get the maximum protection, however this involves trusting your dwarves to dress themselves properly.
The theoretical maximum involves wearing 9 cloaks, I won't list the full outfit in this answer; you can see the details over on the wiki here.
In fortress mode, there is currently no way to control what your civilian dwarves wear, however you can define a custom uniform for your military that allows you to assign multiple of the same of armour or clothing. Assigning this uniform to a squad will get those dwarves to attempt to put on all the items if they are available in your fortess.
There are still some bugs around this area, so instructing your dwarves to wear all this armour may result in them getting stuck picking up equipment.
Having said all this, in practice I would say the actual answer to whether it is better wear multiple pieces of clothing depends on how much time you want to be micromanaging your dwarves wardrobes. I find that equipping my dwarves with a basic set of armour is often adequate.
Can only be certain pieces like leather trousers be categorized as
over and a woollen cloak as cover?
The material type of the clothing has no effect on how it may be worn.
Does over or cover get replaced by armor?
You have the ability to ask your military squads to replace clothing with their uniform, or whether they should wear their uniform on top of their normal clothes. I recommend reading the section on the wiki regarding equipping your soldiers on this.
I never seen clothing wear off, but i have read it does. Is this implemented and how fast does clothing wear off?
It does indeed wear off. You will start to see your dwarves clothes becoming tattered around the second or third year of your fortress, and the clothes start to wear away completely around the third or fourth year of your fortress.
If there are new clothes available, your dwarves will get new clothing when it starts to get tattered.
Do some materials influence the dwarfs mood when walking in certain weather types?
Depending on a dwarf's preferences (if he/she likes a certain material type or a specific type of clothing) that that individual dwarf may get happy thoughts from wearing a piece of clothing or armour of that type. However, this is unrelated to weather conditions.