I have a "matrix" of three indices like $a_{kmn}$ for all $k=1,\ldots,K$, $m=1,\ldots,M$, and $n=1,\ldots,N$ (I think it is called a tensor of order 3, right?). I denote this "matrix" by the boldface letter $\mathbf{A}=[a_{kmn}]$. When I fix one "dimension" (say $k$), I write it as $\mathbf{A}_k$ which is now the $k$th matrix of size $M\times N$.
My question is: are my notations fine? Or are there any standard notations used?
Notations are... notations. Just that. If they work for you and who reads you (if it were the case), they are ok.
Going to the question of standards, I guess the name "tensor" means more than just an array (a 3D array in this case): a tensor of rank three actually can have covariant and contra-variant dimensions and that has something to do with it's physical (or mathematical) meaning and transformation rules under a change of coordinates; this has indeed consequences for the notation ($T_k^{ij}$, $T_{jk}^{i}$, $T^{ijk}$, etc.). I would save the word "tensor" to use it restrictively, depending on the context, and I would rely on the term "array" (three dimensional array, in this case) for the general case.
You can say that $A$ has components $a_{ijk}$ and that $A\in \mathbb R^{m\times n \times p}$, for instance, but the notation $A_k$ for the matrix $$B\in\mathbb R^{m\times n}$$ with entries $$b_{ij}=a_{ijk},$$ I find it ambiguous, since $A_k$ or $A_i$ would always mean that you're fixing the same index. This shortcoming becames evident when you use a number, say $A_3$: whith your notation this is the matrix of componentes $a_{ij3}$, $1\le i\le m$, $1\le j \le n$ and there's no way to refer to a matrix of components $a_{i3k}$ ($1\le i\le m$, $1\le k\le p$) or the analogous version for fixed $i=3$.
Maybe you can try instead of $A_k$, something like $A_{\cdot\cdot k}$, and so: in that case $A_{\cdot \cdot 3}$ and $A_{\cdot 3 \cdot}$, for instance, would not be in general the same matrix (actually they might well be of different sizes.
Even more, you can use $A_{i\cdot k}$ or $A_{ij\cdot}$ to get the one dimensional arrays (vectors) corresponding to fixing two indexes.
And this can actually be easily extended to $n$-dimensional arrays for arbitrary $n\in \mathbb N$.
NOTE: With this notation, you have $$A=A_{\cdot\cdot\cdot},$$ and of course usually you will just type $A$. Also, $$A_{ijk}=a_{ijk}$$ (or $$A_{ijk}=\big( a_{ijk}\big),$$ etc.), that is, a $1\times 1$ matrix, a $1$-dimensional vector or—much better— a $0$-dimensional array). That means that you need not make a difference between uppercase arrays and lowercase components (but it's not a crime either if useful).
FURTHER COMMENT on the 'inspiration' for this notation.
An situation where three or more indexes are needed arises in an experiment where measurements of a magnitude are taken for all combinations of the categories of classification of two (or more) properties (or factors, as they are called in experimental design). Supose that for every combination $(i,j)$ of the first and second factor (if we think of just two factors) there is more than one measurement and the number of repetitions is the same—say $K$—in each case: this is called a balanced design.
That is, we take the measurement $x_{ijk}$ which corresponds to the category $i$ for the first factor and $j$ for the second factor of classification, and it is the $k$-th repetition for that particular combination. For a balanced design of $K$ measurements for each combination of the $I$ categories in factor one and the $J$ categories in factor two, we get the $I\cdot J \cdot K$ measurements $$x_{ijk}, \quad 1\le i\le I,\; 1\le j\le J,\; 1\le k\le K,$$ each is of which is seen as a realization of the random variable $X_{ijk}$.
A simple example: there 6 groups of 10 people, each corresponding to the six combinations of the age factor—with categories 'adolescent', 'adult' and 'elder'— and the treatment factor ('placebo' vs. 'actual medication'), and each one's arterial pressure is measured an hour after taking the (actual or pretended) medication. In this case you have $I=3$, $J=2$ (or viceversa) and $K=10$. And $x_{2,1,5}$ would be the arterial pressure as measured from the fifth individual in the group of adults who are having the placebo.
In the standard model with two factors for the ANalysis Of VAriance (two-way ANOVA) the statistical model is $$X_{ijk}\sim N(\mu_{ij},\sigma^2),$$ all variables being independent. This can also be written $$X_{ijk}=\mu_{ij}+\varepsilon_{ijk},\quad \varepsilon_{ijk}\sim N(0,\sigma^2),$$ and the $\varepsilon_{ijk}$ variables are independent.
In general, there are further details specifying the structure of the matrix $M=\big(\mu_{ij}\big)$, such as the additive model $$\mu_{ij}=\mu+\alpha_i+\beta_j, \quad \sum_{i=1}^I \alpha_i=\sum_{j=1}^J \beta_j=0,$$ a model allowing for generic interactions such as $$\mu_{ij}=\mu+\alpha_i+\beta_j+\gamma_{ij}, \quad \sum_{i=1}^I \alpha_i=\sum_{j=1}^J \beta_j=\sum_{i=1}^I \sum_{j=1}^J\gamma_{ij}=0,$$ which can be simplified to a multiplicative model $$\mu_{ij}=\mu+\alpha_i+\beta_j+\lambda \alpha_i \beta_j, \quad \sum_{i=1}^I \alpha_i=\sum_{j=1}^J \beta_j=0,$$ etc.
So the data and the random variables $X_{ijk}$ form indeed a 3D-array, and when considering quantities such as sums ($S$), means ($\bar X$) or number of data ($n$), the convention of the dots (or sometimes another sign instead, such as $+$ for instance), is standard notation, as in: $$n_{ij\cdot}\:\colon\: \text{number of data in the 'cell' or combination $(i,j)$ of factors}$$ (this is $K$ in a balanced design, but otherwise could vary among different $(i,j)$ combinations); $$n_{i\cdot\cdot}\:\colon\: \text{number of data in the $i$-th category of the first factor of classification},$$ $$S_{ij\cdot}=\sum_{k=1}^K x_{ijk},$$ $$S_{\cdot j\cdot}=\sum_{i=1}^I\sum_{k=1}^K x_{ijk},$$ $$\bar x_{\cdot j\cdot}=\frac1{n_{\cdot j \cdot}}S_{\cdot j\cdot},$$ $$\bar x_{ij\cdot}=\frac1{n_{ij\cdot}}S_{ij\cdot},$$ $$\bar x_{\cdot \cdot \cdot}=\frac1{n_{\cdot\cdot\cdot}}S_{\cdot\cdot\cdot},$$ and so on (the last one is sometimes called the 'great mean').
I hope all the blah helps you get a taste of how this notation works.