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By the definition; all the even numbers have $0$ as a hash value and odd numbers have $1$ as a hash value. So, there is no uniqueness. But finding another one  ,a collison, must be computationally infeasible.

Hash functions are by design are not invertible funtions as permutations. They achive this by2

By the definition; all the even numbers have $0$ as a hash value and odd numbers have $1$ as a hash value.

Another way to see this is the pigeonhole principle. The input size is larger than the hash size, Therefore there exist at least one hash value contains more than one message.

So, there is no uniqueness. But finding another one, a collision, must be computationally infeasible.

Hash functions are by design are not invertible functions as permutations. They achive this by2

1. a) As far as I understand, hashes are just long alphanumeric strings.

In the designs of hash functions is it required that finding a pre,second-image and collision must be computationally infeasible. But there is always a negligible chance of the attacker to find one.

Hash functions are by design are not invertible as permutations. The attacker must find either a preimage or secondary-preimage.

  A powerful entity can search all possible inputs to match the given hash. These examples rainbow table,hashcat may be not as powerful as you imagine but they are on the edge of computing.

Anyone afraid of quantum hashcollision algorithms already has much more to fear from non-quantum hash-collision algorithms.

Nothing except the hardness of finding a collision. If somehow the attackers are able to find a collision as in the malicious valid PDF creating in SHA-1, they can execute it.

0. How do Hashes really ensure uniqueness?

As David gave his answer, no they don't' ensure uniqueness. To see this consider a simple hash (imitating the only compression);

$$H':\{0,1\}^{20} \rightarrow \{0,1\}^{1}$$ $$x \mapsto x \pmod 2$$

By the definition; all the even numbers have $0$ as a hash value and odd numbers have $1$ as a hash value. So, there is no uniqueness. But finding another one ,a collison, must be computationally infeasible.

1. a) As far as I understand, hashes are just long alphanumeric strings.

In the designs of hash functions is it required that finding a pre,second-image and collision must be computationally infeasible. But there is always a negligible chance of the attacker to find one, as in the MD4 case.

Hash functions are by design are not invertible funtions as permutations. They achive this by2

• Bit dependency: each bit of the output is dependent of the every bit of input.
• Avalanching : a single bit change in the input must change $\approx$ half of the bits randomly.
• Non-linearity: prevent from attacking linear systems solving techniques.

The attacker must find either a preimage or secondary-preimage. A powerful entity can search all possible inputs to match the given hash. These examples rainbow table,hashcat may be not as powerful as you imagine but they are on the edge of computing.

Anyone afraid of quantum hash collision algorithms already has much more to fear from non-quantum hash-collision algorithms.

The quantum computers reduced the complexity of hash collision from $2^{b/2}$ to $2^{b/3}$. The non-quantum computers already achieved $2^{b/3}$ with smaller time, The Rho Machine.^1,2

Nothing except the hardness of finding a collision. If somehow an attacker is able to find a collision they can execute it. As recently, an identical-prefix collision attack for SHA-1 performed in PDF files to create malicious valid PDFs.

What you said is called a hash collision. By definition of hash, it is inevitable but the finding one must be computationally infeasible. But if your hash function is considered as weak or a new attack occurs you must change it as for MD5 or SHA1.

Picking one at random, you will have $1/2^{512}$ probability to match the hash as long as the hash function behaves randomly. There is an interesting random hash collision on RC4 on e-mule.

Nothing except the hardness of finding a collision. If somehow the attackers are able to find a collision as in the malicious valid PDF creating in SHA1, they can execute it.

What you said is called a hash collision. By definition of hash, it is inevitable but the finding one must be computationally infeasible. But if your hash function is considered as weak or a new attack occurs you must change it as for MD5 or SHA-1.

Picking one at random, you will have $1/2^{512}$ probability to match the hash as long as the hash function behaves randomly. There is an interesting random hash collision on MD4 on e-mule.

Nothing except the hardness of finding a collision. If somehow the attackers are able to find a collision as in the malicious valid PDF creating in SHA-1, they can execute it.