(Transverse) Ising Model Higher Than Four Dimensions First question: 
Wiki says Ising Model higher than four dimensions can be described by mean field theory. What is the reason for this? Does this mean there is no phase transition for higher dimensions than four? I can see fluctuations become less important in higher dimensions since there are many "neighbors" to stabilize the interactions but I'm looking for a more quantitative argument. 
Second question: 
The transverse Ising model 
$$H = -J\sum_{ij} \sigma^z_i \sigma^z_j-h \sum_i\sigma^x_i$$
is the quantum correspondence of Ising model. Does the statement above hold for this model? 
 A: The criterion for determining whether mean-field theory is good or not is the Ginzburg criterion. You can estimate how accurate mean-field theory is by computing the leading corrections to it. In particular, the exact Ising model is
$$
H = -J \sum_{\langle i j \rangle} s_i s_j,
$$
and mean-field theory is done by taking
$$
s_i s_j = M^2 + M (s_i + s_j) + (s_i - M) (s_j - M),
$$
where $M = \langle s_i \rangle$, and then approximating the last term as zero,
$$
(s_i - M) (s_j - M) \approx 0.
$$
One way to determine that mean-field theory is failing is to calculate the expectation value of this omitted term within mean-field theory and compare it to the magnitude to $M^2$ near the critical point. If that ratio is large, then the term you omitted will cause a large correction, so you're not justified in omitting it. One can show that if the system has correlation length $\xi$, then
$$
\langle (s_i - M) (s_j - M) \rangle \propto \xi^{2-d}.
$$
Meanwhile, within mean-field theory, $M^2 \propto \xi^{-2}$, so the Ginzburg criterion says that mean-field theory is accurate if
$$
\xi^{2-d} \ll \xi^{-2}.
$$
This means that, close enough to the phase transition ($\xi$ large enough), mean-field theory is asymptotically correct for $d>4$ and incorrect for $d<4$. (As an aside, if we considered longer-range interactions, the range of the interaction alters this criterion.)

Does this mean there is no phase transition for higher dimensions than four?

No, this means that there is always a phase transition for $d>4$ (mean-field theory predicts a phase transition).

Second question: The transverse Ising model [...] is the quantum correspondence of Ising model. Does the statement above hold for this model?

The $d$-dimensional transverse-field Ising model (TFIM) actually corresponds to a $d+1$-dimensional classical finite-temperature Ising model. So the upper-critical dimension is $3$ for the TFIM.
