I'm trying to give an answer on a very basic level.
- Digital signatures: You use your private key for "encryption" and they use your public for "decryption". ...
This is only true for one specific digital signatre (RSA signature), and simply not true in general. This is like saying "Every animal lives in the water" - also not true in general. Even for RSA, we have to take a step back:
This symmetry is only true if we consider the basic textbook-RSA. However, this is not secure for either encryption or signatures. The textbook variant is basically just the generation of the keys and to show how the trapdoor works. For actual encryption and signatures you need to use a so called "padding scheme" - and there are different padding schemes (e.g. RSA-OAEP for encryption, RSA-PSS for signatures). Exchanging those is either not possible or has no guarantee for security any more.
.2 ... Now I have the impression that private and public key are symmetric in a sense that you can choose one for encryption and then the other one is for decryption but you could also do it the other way round. ...
Again, this is just for the RSA systems, not in general. And the short answer is: Don't do it, even if you are not aware of the reason.
With a bit more detail: The algorithm for key generation is just that - it does not include the actual reasons for every decision. The algorithms are written in a way that they work - if you follow the instructions precisely. The algorithms should be usable without extensive knowledge of cryptography, but the exact reasons why something was chosen with certain properties usually does require that knowledge.
Here is an example: If you choose $d$ small, then calculate $e$ instead and publish $(N,e)$ as public key as usual, then the system is still working correctly and the steps are basically the same as the regular key generation. However, it is not secure at all, because there is an efficient attack against small values for $d$ (see Wiener's attack). This is just one example how you can screw up really badly, if you don't know exactly what you're doing.
... Where is the math with trapdoor function used here if it is in a sense completely symmetric?
The trapdoor is hidden in knowing both $e$ and $d$ at the same time. With that, you can also find the factorization of $N$, but the explanation is quite advanced. Bascially it is like a two piece puzzle, and you need both pieces to make sense of it. And since of the pieces is known to everyone, you have to keep the other hidden.
- Is it possible to easily derive the public key from a private key? I have heard that in theory NO but in practice YES...
We have the basic assumption that public keys are public: Everyone knows them, no exceptions. But let's assume the public key isn't known. In textbook RSA, if you only know $N,d$ and know nothing about $e$ (and $e$ is as large as $N$), then you can't find $e$. In other systems, e.g. ElGamal encryption, the private key is an exponent, and if you know that and the parameters of the modular group, then calculating the public key is easy.
You referred to a difference in theory and practice here, which is easy to explain: Usually the "private key" in practical systems is not just the very minimum information required. More often it either has the public key in there as well, or you can easily derive the public key from it. For example in RSA, you would call $(N,e)$ the public key, and keep $(N,e,d,p,q)$ as private key. Sure, there is some redundant information, but even with $N,e,d$ with up to $2048$ bit and $p,q$ with $1024$ bit, we're only at $1024$ Byte in size.