What are Quantum Groups?

This answer is a small attempt to address your third question. You may find the article,

Jethro van Ekeren, The six-vertex model, $R$-matrices, and quantum groups,

to be helpful.

One of the main sources in the development of quantum groups was the field of exactly solvable models in statistical mechanics. A number of simple mathematical models were devised at the same time as quantum mechanics was being developed in order to understand phase transitions (changes of state) that occur in certain magnetic materials like iron, and that can also be used to understand properties of more familiar changes of state like the liquid-gas transition. One of these models was the Ising model, which is a classical (non-quantum) model. Similar models that came much later were the six-vertex and eight-vertex models. On the quantum mechanical side, there was the Heisenberg spin chain, among others.

The remarkable thing about these models is that, at least for certain values of their parameters, it is possible to compute certain physically interesting quantities, such as the free energy, exactly in the limit of infinite system size (the thermodynamic limit). In physics, symmetries of a system are associated with conservation laws. What makes these particular systems exactly solvable is that, in the thermodynamic limit, they have an infinite-dimensional group of symmetries, and therefore infinitely many conservation laws. (Without these conservation laws, the computation of the free energy become increasingly intractable as the system size grows.) These infinite-dimensional symmetry groups are mathematically interesting. In 1944 Lars Onsager computed the free energy of the two-dimensional Ising model with the external magnetic field parameter set equal to $0$ by introducing a certain infinite-dimensional algebra which was later discovered to be connected to Kac–Moody algebras.

Onsager's solution was considered by physicists to be difficult, and a number of alternative solution methods were discovered in subsequent decades. One of these—the method of commuting transfer matrices—was particularly fruitful in terms of generalizations. The Ising model is defined on a lattice—a two-dimensional square lattice—in Onsager's work. (This is intended as a simplified model of the crystalline lattice of real metals.) There is a binary (two-valued) variable (a "spin") associated with each lattice site, and the spins experience interactions with their nearest neighbors. The transfer matrix is an operator that corresponds to adding a row of sites to the lattice. In the Ising model, six-vertex model, and other exactly solvable two-dimensional models, the key to solvability is that transfer matrices with different values of a certain parameter commute with each other. The quantum mechanical models mentioned above, such as the Heisenberg spin chain, are one-dimensional models with Hamiltonian (energy operator) given by a matrix in the same commuting family. The members of this commuting family can be thought of as physical operators that represent conserved quantities of the spin chain (since they commute with the Hamiltonian, these quantities do not change with time).

Rodney Baxter discovered that commutativity of transfer matrices is implied by what is now known as the Yang–Baxter equation, which involves an operator called the $R$-matrix. The $R$-matrix can be regarded as the fundamental building block out of which transfer matrices are constructed, corresponding to the interactions of a single lattice site with its neighbors. The Yang–Baxter equation relates two different ways in which three sites can interact. It has a graphical representation closely connected with knot theory.

Quantum groups arise as algebraic structures in which $R$-matrices satisfying the Yang–Baxter equation naturally arise. These solutions to the Yang–Baxter equation give rise to new families of commuting transfer matrices, and therefore to new exactly solvable models describing new types of phase transitions.

There are connections between these models and other parts of physics. These particular exactly solvable models have critical points, at which the physical system exhibits a form of scale invariance. Scale invariance actually implies the stronger conformal invariance, and these models at their critical points have continuum limits called conformal field theories. These are a key ingredient in string theory, and play a role in many interesting mathematical developments as well (for example in the proof of the Moonshine conjectures).

Ok, a couple of months have passed since I posted this question, and I have begun this project on quantum groups. I am now in the condition of answering a couple of the questions I posed.

1. There are many different definitions of quantum groups, all of which are related somehow. A possible definition is: a (Drinfeld-Jimbo) quantum group is a $1$-parameter deformation of the universal enveloping algebra $U\mathfrak{g}$ of a Lie algebra $\mathfrak{g}$ which is always a Hopf algebra.
2. The subject is interesting because it relates to the subject of quantum integrable systems in quantum mechanics. There is an ongoing effort trying to geometrize the notion of quantum groups (as stated at page $28$ of the paper of Maulik and Okounkov).
3. See the wonderful and exhaustive answer by Will Orrick.
4. To learn quantum groups you will need some knowledge of physics (to have a motivation for the subject and maybe to get some intuition), Lie groups and Lie algebras, and representation theory.

For the questions added in the edits:

1. I have just begun to read the paper, maybe in a couple of months, when I'll have a clearer idea of what is happening, I will update this answer to give an overview. From what I have been told by my professor, it should be about finding a geometrical approach to quantum groups.
2. To understand the paper easily you will need a lot of algebraic geometry (you should at least know what a Nakajima variety is, and know stuff about Gromov-Witten invariants) and at least some K-theory. I have to admit I am struggling, but I am also learning a lot.

Edit: There are a few more subjects you will need to know to approach the paper and hope to undertand the majority of it. I would advise to take a look at:

• Symplectic geometry (for example here, or any of the other references you can easily find online), in particular you should know at least what a moment map is.
• Geometric Invariant Theory (GIT), which you can look up here.
• Equivariant cohomology, if you have access to Springer through an university, Bott's introduction is really clear, else you can probably find something online.

I will update if I find other useful stuff.