Notation to work with vector-valued differential forms

What it the standard notation used while working with vector-valued differential forms?

I tried using abstract index notation, for example denoting a $1$-form valued $2$-form as $P_{i[bc]}$, but I'm not sure I'm doing it right.

What is the standard way to denote vector-valued differential forms (at least of kind $\Lambda^p(T^*M) \otimes \Lambda^n(T^*M)$), tensor and exterior products, exterior derivative, Hodge dual, and contractions of all kinds?

If you are interested why do I need all these, that's because of fluxes and balance equations, and not only for scalar quantities.

Abstract indices is a good alternative to work with, just don't be afraid of using different ranges of symbols dedicated to each bundle you deal with. For instance, I prefer to use the letters $a$, $b$, $c$ etc (the initial segment of the Latin alphabet) as the labels for the tangent and cotangent bundles (depending on the position), and juxtapositions of these letters denote various tensor bundles. Using parentheses and brackets we can indicate tensor parts of these bundles.

Thus, as you correctly noticed $E_{a[bc]}$ can be seen as the bundle of 2-form-valued 1-forms, and this notation is indeed standard. If there is another bundle that I want to be treated differently from the tensor bundles, I would use letters $\Phi$, $\Psi$, $\Xi$ etc associated to this bundle, and again tensor products of thus bundle and its dual will be simply denoted by using letter $E$ with the letters from the specified range attached as indices in various positions. We may have something like $E^{\Phi \Psi}{}_{\Xi}$ or $E^{[\Phi \Psi]}{}_{\Xi}$, and so on. The question is how to introduce the usual operations in this notation. This has been already done. See the references below.

For instance, the exterior covariant derivative of a 1-form with values in 2-forms will be written as simple as $2\,\nabla_{[i}P_{j][bc]}$. The notation $P_{[a}Q_{b]}$ can be used for the wedge product $P\wedge Q$ (up to the coefficient). A connection in a vector bundle $E^{\Phi}$ will be then a map $\nabla_{a}\colon E^{\Phi}\to E^{\Phi}{}_{a}$ admitting a certain sloppiness of the language. Another useful trick with the abstract indices is not always to use the indices ("suppressing") which comes naturally when one calculates in this notation a lot. For the Hodge dual you may probably want to use the volume form $\varepsilon_{a^{1}\dots a^{n}}$ which is a $n$-form with certain properties.

References:

1. R.Penrose, W.Rindler "Spinors and space-time. Vol.1" which is usually referred to as the origin of the abstract index notation (in fact, it was known to physicists a while before the book appeared, but R.Penrose gave a logical foundation to expose its mathematical rigor in full glory).
2. R.Wald "General Relativity", see Appendix B where one can find useful formulae for differential forms given in the abstract index notation. This book is somewhat more accessible then the previous one.
3. R.W.R.Darling "Differential Forms and Connections" is more suitable for mathematicians, and is a useful reading as well. The author uses the classical "invariant" notation, such as $d^{\nabla}\omega$ for exterior covariant derivatives.

Edit. As requested I attempt to apply the abstract index notation to show the Leibniz rule for the case of a 1-form $v_{a}$ and a 2-form $w_{ab} = w_{[ab]}$. So, we want to show $$ (d(v \wedge w))_{abcd} = (dv \wedge w)_{abcd} + (-1)^{deg(v)}(v \wedge dw)_{abcd} $$ The definitions disentangled give $$ (v \wedge w)_{bcd} = \frac{3!}{1! \cdot 2!} v_{[b} w_{cd]} $$

$$ (d(v \wedge w))_{abcd} = 4 \nabla_{[a} (v \wedge w)_{bcd]} = \frac{4!}{1! \cdot 2!} \nabla_{[a} (v_{b} w_{cd]}) \tag{1} $$

$$ dv_{ab}=2 \nabla_{[a}v_{b]} $$

$$ (dv \wedge w)_{abcd} = \frac{4!}{2!\cdot 2!} (2 \nabla_{[a}v_{b})w_{cd]} = \frac{4!}{1!\cdot 2!} (\nabla_{[a}v_{b})w_{cd]} \tag{2} $$

$$ (dw)_{bcd}=3 \nabla_{[b} w_{cd]} $$

$$ (v \wedge dw)_{abcd} = \frac{4!}{1! \cdot 3!} (3 v_{[a} \nabla_{b} w_{cd]}) = \frac{4!}{1! \cdot 2!} ( v_{[a} \nabla_{b} w_{cd]}) \tag{3} $$

It remains to compare (1) with (2) and (3) to understand what is going on: the alternation is responsible for the sign $(-1)^{deg(v)}$ which is just $-1$ in our case.

Remark. If our forms have coefficients in vector bundles we need to take care to define what the wedge product really means for the calculations to make sense.

Let $E$ be a smooth rank $k$ vector bundle on a smooth manifold $X$. I would write an $E$-valued $p$-form (an element of $\Omega^P(E) := \Gamma(X, \bigwedge^P(T^*X)\otimes E)$) locally (i.e. on a trivialising open set for the bundle $\bigwedge^P(T^*X)\otimes E$) as $$f_I^{\ \alpha}dx^I\otimes s_{\alpha}$$ where $I$ is a $p$-multiindex and $\{s_{\alpha}\ |\ \alpha = 1,\ \dots,\ k\}$ is a basis of local smooth sections of $E$. Note, I am using Einstein summation notation.

As for extending the operations on forms, sometimes authors use the same notation, other times they will include the name of the bundle using either a superscript or a subscript. For example, the Hodge star acting on $E$-valued forms is sometimes written as $\ast_E$.