I believe you will find such examples for $X=\mathbb{C}$ and $T$ a rational map in
Mary Rees, Ergodic rational maps with dense critical point forward orbit, Ergodic Theory and Dynamical Systems 4 (1984), 311-322. official version.
In my Ph.D. thesis, I showed that some of these even support a metric with respect to which these dynamical systems are ``hyperbolic''. This metric gives a notion of length of curves comparable to the usual metric on the Riemann sphere, but is defined by a function which is singular on a dense set of points on the sphere (the forward orbit of the critical point).
Answer to the quick version. Yes it is true as soon as $(X,\mu)$ is a Lebesgue space. Beware that the transformation on the product $A_i\times B_i$ is not necessarily a true product, but instead it is a skew-product of the form $(x,y)\mapsto (T_x(y),y)$. This follows from the ergodic decomposition theorem, together with the classification of measurable partitions.
Recall that if T is an invertible measurable transformation acting on a Lebesgue space X, then there is a measurable partition $C_i$ (which may have uncountably many elements) and
probability measures $\mu_i$ on $C_i$ such that all $C_i$ are invariant by T, T is ergodic w.r.t $\mu_i$ and $\mu$ is obtained by integrating the $\mu_i$.
$$\mu(A) = \int_X \mu_i(A) d\mu$$
There are two kinds of ergodic components $C_i$. The one of positive measure, there are at most countably many such components. Let us remove these components from $X$, together with the periodic points, which are easily dealt with. Rohlin structure theorem on measurable partitions (1947) now says that
there is a isomorphism between $([0,1]\times [0,1], \lambda)$ and $(X,\mu)$ such that the pullback of the measurable partition $(C_i)$ is the decomposition into horizontal lines $([0,1]\times \{i\})_{i\in [0,1]}$. A reference is the book of Parry, "entropy generators in ergodic theory".
Here is how the ergodic decomposition is often used. If it happens that a result is true for an ergodic transform, then it is true for an arbitrary transform in restriction to its ergodic components, and you (may) recover the result on the whole space $X$ just by using the integral formula given above.
A reference for the ergodic decomposition for countable groups action is Glasner, "ergodic theory via joinings" th. 3.22.
Finally the result you are alluding in your last question is a section theorem. Given a measure preserving transform between two Lebesgue spaces X and Y, there is a section from Y to X, up to null sets, and some warning is in order here because this is not true in the Borel category. I think this is again due to Rohlin, and it can be deduced from its structure theorem for measurable partitions. Have a look at the book of Parry, but really this is overkill.
EDIT: following the comments of R.W., here is a counterexample to having a true product, instead of just a skew-product. On $[0,1]\times [0,1]$ take
$(x,y)\mapsto (x+y\ \ mod\ \ 1,y)$, together with Lebesgue measure. The restrictions to the fibers $[0,1]\times \{y\}$ are ergodic for a.e. y, and give uncountably many different isomorphic systems, as can be shown by looking at their spectra.
Best Answer
Edit: I've updated this answer to reflect the helpful comments made by Andres Koropecki and Ian Morris.
As the other answers mentioned, the first crucial distinction you must make is that some properties refer to a topological dynamical system $(X,T)$, while others refer to a measure-preserving dynamical system $(X,T,\mu)$. Thus there are two different sets of definitions. Let me attempt a sketch at some of the relationships within each set.
First suppose you have a topological dynamical system $(X,T)$. Then four of the key properties are topological transitivity, topological mixing, minimality, and unique ergodicity. The first three are related by
Unique ergodicity is independent of those three properties. The picture is the following.
Counterexamples 1-9 illustrating the strict containments are as follows. (These may not be the simplest or the earliest counterexamples in each case, and I welcome corrections or improvements. This is based on some quick googling for things not already in my memory, plus the helpful additions offered by commenters.)
1. $X = \Sigma_2 \times \{a,b\}$, the direct product of a full two-shift with a period-two orbit, where the dynamics is $\sigma\times S$, with $\sigma$ the shift map and $S$ the map interchanging $a$ and $b$.
2. $X=\Sigma_2$.
3. Constructed by Bassam Fayad, Topologically mixing and minimal but not ergodic, analytic transformation on $\mathbb{T}^5$, 2000.
4. Constructed by Furstenberg, Strict ergodicity and transformation of the torus, 1961.
5. An irrational flow on the torus, slowed down near a single point: see the comment below by Andres Koropecki.
6. As Ian Morris points out in the comments, the identity map on a singleton set works here. A less trivial example was given by Karl Petersen, A topologically strongly mixing symbolic minimal set, 1970.
7. Rotation of the circle by an irrational angle.
8. Direct product of the example from 5 with a periodic orbit. (Again as suggested by Andres in the comments.)
9. North-south map: a map $T\colon [0,1]\to [0,1]$ with fixed points at $0,1$ and such that $T(x) < x$ for all $x\in (0,1)$. Identify the endpoints $0$ and $1$ so that this is a uniquely ergodic circle map.
A couple things are probably worth pointing out.
All of the above is for topological dynamical systems, where no invariant measure is specified. Then there are the ergodic properties: those that depend on a system preserving an measure $\mu$. For these one has the ergodic hierarchy.
It is very often the case that one wishes to study a topological dynamical system as a measure-preserving system by equipping it with an invariant measure, and in this case it is quite reasonable to ask about the relationships between the two different classes of properties. But this depends on which invariant measure you choose, because in general there may be very many of them. One may ask what properties of $(X,T)$ let you pick invariant measures $\mu$ with certain nice properties, and this is a whole different story which would expand this answer far beyond the bounds of propriety.