Difference between QFT and quantum mechanics It is often said Quantum mechanics is $d=0$ and QFT can have $d = \infty$. What does this mean and how does it makes tunneling possible in Quantum mechanics and not in Quantum Field Theory? What is $d$ exactly?
 A: There are a few things to unpack. I am going to intentionally avoid using the notation $d$ for the quantities I define, except to explain later the connection to what I think you might mean by $d$ and some various notions I will define.
Spacetime dimension $D$
First, physics takes place in space-time with a space-time dimension $D$. Often we break this down into the number of spatial dimensions (which I will call $d_s$) and the number of time dimensions ($d_t$), so $D=d_s+d_t$. Almost always, unless you are doing something quite specialized, we study theories with $d_t=1$, so we often write $D=d_s+1$.
For example: our everyday world has 3 spatial dimensions and one time dimension, so we the Standard Model describing our world lives in $D=3+1$. Superstring theory lives in $9+1$ dimensions (at least... perturbatively...).
The number of dimensions does not depend on whether the theory is quantum or classical, or (if quantum) whether the theory is a field theory or a theory of one or more particles.
Number of degrees of freedom $N$
Second, a given theory has a certain number of degrees of freedom $N$. There are different ways you can count the degrees of freedom, but I will take a fairly standard approach and say that $N$ is equal to the number of real numbers you need to specify as initial data in order to solve the dynamical equations of motion.
Particles
For a particle moving in one spatial dimension (for example: a ball you through straight up in the air, or a ball rolling on a table) $N=2$. The reason is that you need to specify the initial position and velocity of the particle. Note this particle is moving in $D=d_s+1=1+1$ dimensions.
For a theory with $1$ particle moving in $3+1$ dimensions, there are $N=6$ degrees of freedom (one position and one velocity component along each of the 3 spatial dimensions).
For $10$ particles in $D=3+1$ dimensions, $N=60$. For $p$ particles in $D=d_s+1$ dimensions, there are $N=2 p d_s$  degrees of freedom. This formula is true for classical and quantum theories, but we are talking about particles and not fields.
Fields
Now we come to fields. The simplest case is a scalar field. A scalar field takes one value at every point in space for each time.
Of course, the number of points in space is infinite (at least if $d_s>0$), so $N=\infty$, for any value of $D$. But we can talk about the the number of degrees of freedom per point in space, which I will call $n$. I will refer to $n$ as the number density of degrees of freedom, although this is not standard nomenclature (I don't know if this quantity has a standard name or not). The number density of DOFs is $n = N/V_s$ (where $V_s$ is the volume of all space at a fixed point in time).
To specify the dynamics of a scalar field, we need to specify the value of the scalar field (similar to the position of the particle) and the time derivative of the scalar field (like the velocity), so there are two degrees of freedom at every spatial point, and therefore $n=2$.
It is also possible to have more complicated fields. For example, we have have a vector field where we specify a $D$ dimensional vector at every point in spacetime, or a scalar field theory with $k$ scalar fields. For the vector field, $n=2D$, and for the scalar fields, $n=2k$. In all cases, $N=\infty$ and $D=d_s+1$, and we haven't specified $d_s$.
The value of $N$ does not depend on whether the theory is classical or quantum, but we are talking about a field theory and not a particle theory.
An interesting trick: particles as 0+1 dimensional fields
A very interesting observation is that we can think of a particle theory in $D=d_s+1$ dimensions as a field theory with $d_s$ scalar fields in $D=0+1$ dimensions. This is a bit of a singular case of a field theory, since $V=0$.  Since $D=0$, $N$ is finite, unlike in the case where $d_s>0$. There is one spatial point, and there are $d_s$ fields we need to specify $N=2d_s$ pieces of initial data to evolve this system. But with that in mind, note
(a) the number of degrees of freedom $N$ for a particle in $D=d_s+1$ dimensions is $N=2d_s$, which matches the number of degrees of freedom $N=2d_s$ for $d_s$ scalar fields in $D=0+1$ dimensions, so it makes sense to identify these theories, and
(b) the reason this trick is useful, is that it lets us apply some tools and intuition from field theory to a particle theory. However it's just a different point of view, and not necessary to view particles in this way.
This observation is independent of whether the theory is quantum or classical.
Quantum tunneling
Finally, we come to quantum tunneling. Of course, this phenomenon only happens in quantum, and not classical theories.
The rate of tunneling processes depends on the number of degrees of freedom, $N$. If we have one particle ($N=2d_s$), then there is a certain probability for tunneling to occur, call it $p$, and of course $p<1$. If we have two particles ($N=4d_s$), then the probability that both will tunnel is, roughly, $p^2<p$. For a field theory, since $N=\infty$, the probability that all the degrees of freedom in the field will simultaneously tunnel, vanishes (since $p^N$ becomes exponentially small for large $N$).
Now, the ground state of a quantum particle in a double-well potential is a superposition of a state with the particle localized in one minimum, and a state with the particle localized in the other minimum. However, as a direct consequence of the vanishing tunneling probability, for field theory the ground state only has the field localized in one minimum.
However, while it's true that the whole field configuration everywhere in space cannot tunnel (in the sense that it is infinitely unlikely that this will happen), there is still a finite (but very small) probability the field will undergo a tunneling event in a finite region of space. The tunneling event becomes more and more unlikely as the volume of the region increases.
Tunneling events in field theory are calculated using instantons, which are finite-action classical solutions to the Euclidean equations of motion. The probability of the instanton event is proportional to $e^{-S}$, where $S$ is the action of the instanton solution.
This does not contradict the point about the ground state of the field being in one minimum, because if the field does tunnel to a different minimum in a finite region of space, then this "bubble" where the field is in the new minimum will expand and convert the rest of space to this new ground state. (So long as you don't worry about gravity... if you do then the story is more complicated).

Summary
So, putting this all together, here is how I would answer your question.

*

*The quantum mechanics of a single particle in $D=d_s+1$ spacetime dimensions, is equivalent to a quantum field theory with $d_s$ scalar in $D=0+1$ dimensions. If by $d$ you mean the number of spatial dimensions in this quantum field theory, then I agree with your first statement: "Quantum mechanics is =0".

*Quantum field theory has an infinite number of degrees of freedom if there are one or more spatial dimensions (if $d_s>0$ then $N=\infty$). So if by $d$ you mean the number of degrees of freedom in a quantum field theory with $d_s>1$, then I agree with your statement that a QFT can have $d=\infty$.

*As we have discussed, tunneling is possible in both quantum mechanics and quantum field theory, however while the ground state of a quantum mechanical system with a double well potential will involve a superposition of the particle being in the two different minima, in quantum field theory the ground state of the field is simply localized in one minimum. The intuitive reason is that $N=\infty$ for a QFT, so it is infinitely unlikely that all the degrees of freedom will tunnel.

Since I have used $d$ in different ways in the above three bullet points I don't think your sentence is strictly consistent, but that is how I would interpret the elements of your question.
A: The quantum mechanics of a particle in $d$ dimensions is a $0+1$ dimensional QFT with $d$ scalar fields. The quantum mechanics of a string in $d$ dimensions is a $1+1$ dimensional QFT with $d$ scalar fields. As long as a quantum field theory lives in more than $0+1$ dimensions, the configurations quantized at a fixed time are allowed to vary locally. This is the sense in which they have infinitely many more degrees of freedom.
The probability of each position of a particle is described by a wavefunction which sends a number to $\mathbb{C}$. The probability of each shape of a string / membrane / field / whatever is a described by a wavefunction which sends another function to $\mathbb{C}$. I.e. a wavefunctional.
A: To answer your question title, quantum mechanics and quantum field theory respectively quantize classical mechanics and classical field theory. In particular, both field theories make allowance for special relativity.
To answer your question body, $d$ denotes a dimension. (In this sense, "dimension" is an integer, not a thing that integer would count; for example, "$3$-dimensional space" is of dimension $3$, rather than having $3$ dimensions.) QM can be construed as a $0$-dimensional field theory, while QFTs become infinite-dimensional conformal field theories in the high-energy limit.
