For the first question, I think the answer is no. Consider the following example:
$X = Spec k[x,y,z]/(xy - z^2)$ a quadric cone. Consider the Cartier divisor $D = V(z)$. It has two irreducible components corresponding to the ideals $(x,z)$ and $(y,z)$ respectively (these are non-Cartier, Q-Cartier divisors). Both components smooth and they meet at the origin and so the multiplicity of $D$ (of this nodal singularity) is $2$. By multiplicity, I assume you mean the multiplicity of the scheme $D$ at a point.
On the other hand, if you blow up the origin $(x,y,z)$ you get a chart $$k[x/z, y/z, z]/( (x/z)(y/z) - 1).$$ The pull back of $D$ on this chart is just $z = 0$ (one copy of the exceptional divisor) so the order of $\mu^*(D)$ along $E$ is equal to 1. (The order of the components along the exceptional divisor is $1/2$ in each case, but they are $\mathbb{Q}$-Cartier)
There's a deeper problem in your first question though. If I recall correctly, in general, when you blow-up a point $x \in X$ on a singular variety, there isn't a unique prime exceptional divisor lying over $x$. There are probably multiple such divisors. To make matters worse, the pull back of your given Cartier divisor can have different multiplicities along these different exceptional divisors.
For the second question:
You assume that the pair $(X, \Delta)$ is klt, and you define the discrepancy at $E$ to be the order along $E$ of $K_Y - \mu^*(K_X + \Delta)$. Then you say that you know that $a(E, X, \Delta) \leq 1$ if $X$ is smooth. This isn't true.
I assume you know that the definition of klt implies that these discrepancies are all $> -1$. However, consider the following example.
$X = Spec k[x,y,z]$ and $\Delta = 0$. This pair is certainly klt. When you blow up the origin though, the relative canonical divisor $K_{Y/X} = 2E$, two copies of the exceptional (if you blow up the origin in $\mathbb{A}^n$, you get $n-1$ copies of the exceptional divisor). If you blow up points on that exceptional divisor (and repeat), you get further exceptional divisors with greater and greater discrepancy.
Hopefully I didn't misunderstand the question.
You gave the definition of normal crossing divisor. The definition of simple normal crossing is the following.
A Weil divisor $D = \sum_{i}D_i \subset X$ on a smooth variety $X$ of dimension $n$ is simple
normal crossing if any component $D_i$ is smooth and for every point $p \in X$ a local equation of $D$ is $x_1\cdot...\cdot x_r$ for independent local parameters $x_i$
in $O_{p,X}$ with $r\leq n$.
What goes wrong in the case of three line passing through the same point in $\mathbb{P}^2$ is that you need three local parameters in a neighborhood of $p$ and $3 > 2$. The same for the cusp.
More generally, let $D = \sum H_i\subset\mathbb{P}^n$ be a reduced divisor, where the $H_i$'s are hyperplanes. Consider the points $h_i\in\mathbb{P}^{n*}$ dual to the the $H_i$'s. Then $D$ is simple normal crossing if and only if the $h_i$'s are in linear general position in $\mathbb{P}^{n*}$. You see that three lines through the same point in $\mathbb{P}^2$ correspond to three points on the same line in $\mathbb{P}^{2*}$. Therefore this case does not work. More generally, for a curve $C$ on a surface, simple normal crossing means any irreducible component is smooth and $C$ has at most nodes as singularities.
Best Answer
I think at the very least you should require that $d_i \leq 1$ even if this is not clear from the literature.
If you don't require that $d_i \leq 1$, then lots of things don't work.
Problems with the given definition
Let me give an example. Set $X = \mathbb{A}^2$ and let $D = 3 \text{div}(x)$. Then $(X, D)$ is its own log resolution and so using your definition, the $b_j$'s are all zero and so this pair is log canonical. However, if $Y \to X$ is the blowup of the origin, then there is one exceptional divisor $E$ and $E$'s coefficient in $K_Y - f^*(K_X + D)$ is equal to $1-3 = -2$.
If you require that your condition holds for all log resolutions (or even all birational maps $f : Y \to X$ with $Y$ normal) then that implicitly guarantees that the coefficients of $d_i$ are all $\leq 1$. Just blow up the points on $D_i$ with coefficients $> 1$ repeatedly. Note that in Kollár-Mori, they do require all resolutions / valuations.
One really wants to say that $(X, D)$ is log canonical if and only if $(Y, -K_Y + \pi^*(K_X + D))$ is log canonical for any proper birational map $f : Y \to X$ with $Y$ normal. This also guarantees that $d_i \leq 1$.
You need both $d_i \leq 1$ and $d_i \geq 0$ in order to guarantee the Kodaira-type vanishing theorems you rely on. For instance, if $(X, D)$ is LC and if $L$ is a line bundle such that $L-K_X-D$ is ample, then $H^i(X, L) = 0$ for $i > 0$ if your $d_i$ are in $[0, 1]$.
An alternate definition
Let me say that I prefer a slightly different way to setup the definition. Write
$$K_Y = f^*(K_X + D) + \sum_j b_j E_j$$
without subtracting off the strict transform.
First notice that this means that some of the $E_j$ will be non-exceptional (that's totally ok). Then
Definition $(X, D)$ is log canonical if the $b_j \geq -1$. $(X, D)$ is KLT if the $b_j > -1$.
This definition then directly forces the $d_j \leq 1$ (respectively $d_j < 1$ for KLT). Setup this way, everything is completely independent of the choice of resolution.
Note that I didn't require that the $d_j \geq 0$ (this depends on the application, but based on 3. above, it might be reasonable to allow negative $d_i$ in the definitions. I should note that allowing negative $d_i$ is probably slightly more problematic for LC than it is for KLT, but I've written enough already).
DLT and ?LT singularities
Note that DLT singularities are a bit different (and then the $d_i$ are assumed $\leq 1$). Then they are setup not to be independent of the choice of resolution. See the recent book by Kollár and Kovács on singularities for a discussion of what kind of ``resolutions'' DLT are independent of (or think about Szabo's characterization of DLT singularities).
There are other notions of log terminal in the literature too, which make other weaker requirements about requiring things for all or some resolutions (for instance, weakly log terminal).