Quantum key exchange skepticism/confusion
Let’s take your questions in order. Note that I’m a physicist working in quantum cryptography, so my opinion on this might be biased
1. What about authentication ?
The classical channel between Alice and Bob has to be authenticated in order for the protocol to work. Formally, this is a pre-requisite for quantum key distribution (QKD), and is not part of the protocol.
In practice, we often uses information theoretically secure schemes (see this answer for pointers), which consume secret keys. The point is to generate key through QKD faster than the authentication consume it, making the whole system a key expansion system.
2. Wait ! Can’t Eve guess half of the basis (filters) without being detected ?
No ! That’s the whole point. Eve does indeed know half of the key, but she’s detected by Alice and Bob. Eve choice and Bob’s choice are independent, so, Eve will have chosen the wrong filter for half of the photons kept by Alice and Bob, and this wrong filter choice would change the photon polarization, inducing errors in Alice and Bob strings (in this case $\tfrac12\times\tfrac12=25\%$ error.) detected in step 5. of your description of the protocol. This lead us to the next question .
3. What is the maximal error rate ? And why ?
An exact answer to this question is still a research subject, and is obviously protocol dependent, but the same ideas (with different numbers) are essentially valid for all protocols.
The attack you proposed is known as an intercept-resend attack, describing an attack where Eve measures the photons and resends something to Bob depending on her measurement. It is proved that there is no way for Alice and Bob to extract a secret key when such an attack occurs, which corresponds to 25% error rate for BB84 protocol. However, this is only an upper bound on the tolerable error rate, and this attack has no reason to be optimal.
To find a lower bound, one uses different techniques to optimize over the sets of attacks, which also depend on the exact way the final key is extracted (after error correction and privacy amplification, in order to use the lefover hash lemma). A commonly used lower bound is $e=11\%$ which corresponds to a case where Eve gather as much mutual information on Alice's string as Bob, but some research papers have some way to tolerate a higher error rate.
4. We always assume Eve is passively oberving ...
NO ! The whole point of QKD is that she cannot passively observe ! Heisenberg uncertainty principle ensures that any observation is active and disturbs the system. Usually, we assume Eve can do anything, but the tricky part is to properly define “anything”, and optimize over this set.
5. How does the leftover hash lemma actually work
I’m a physicist, and I will not give you a theoretical answer on this. You can read the publications of Renato Renner to have a rigorous analysis of this lemma in the presence of a quantum adversary.
On a practical point, Alice and Bob use error correcting codes (ECC) to correct the errors in order to have the same key. Doing that, they leak some information to Eve (at most the number $c$ of exchanged ECC bits, $c≥h(e)$ for BB84). This leakage is added to amount of information which has leaked during the photon exchange, and which can be evaluated from the error rate $e$ (at most $h(e)$ for BB84). Alice and Bob pass their key through a universal hash function, with an output small enough ($&lt;1-c-h(e)$) to ensure the leftover hash lemma applies.
Bonus question (from the comments) : is QKD relevant ?
I’m an academic, and I find it interesting from a fundamental point of view. It clearly changes the way physicist think about quantum mechanics, and is relevant in this aspects.
Whether it will soon be relevant to use QKD now in an industrial setting is a question of engineering, economics, future prediction and beliefs on what the NSA actually does ...
QKD is more difficult to implement than classical cryptography, because it needs specific hardware and has a limited range. However, it is indeed possible today to make QKD systems working over 200km, and small networks exist across a few cities. Some people work to make it practical and cheaper, and they may succeed.
In my opinion, the only reasonable application of QKD today is on securing expensive data which should stay secret for more than a decade (i.e. on an horizon where technological evolution is difficult to predict). The two properties which make QKD useful in this case are :

Its security relies on something completely independent than classical cryptography and it is easy to combine the two such that both have to be broken in order to break the scheme.
In order to break an imperfect implementation of quantum cryptography (aka quantum hacking), one need to break it at the moment where the photon exchange take place. You cannot record the messages in the hope that a weakness in the protocol will later be  found.