Categorification of coends and ends I describe below a categorified version of the coend construction, "2-coend" for short.  It takes as input a collection of 1-categories $\{W_{xy}\}$ which afford left and right representations of a (weak) 2-category $A$.  The output is a 1-category $C$ which can be thought of as the self-gluing or self-tensor product of $\{W_{xy}\}$ via the left and right $A$ actions.

Questions: Does this construction appear in the literature anywhere (other than here)?  Does the special case where $A$ is replaced by a tensor category (2-category with one object) appear in the literature anywhere?  How about the special case where $\{W_{xy}\}$ is the product of a left $A$ representation and a right $A$ representation?  (So a categorification of tensor product rather than a categorification of the more general coend.)

The Drinfeld double is a special case of the 2-coend, and dually the Drinfeld center is a special case of the 2-end.  (See final remark below.)
The definition I give below seems natural and inevitable to me, so I'm a little surprised I have not (yet) come across it elsewhere.  If this MO question does not turn up any citations, I will tentatively assume that the definition is new.
Final remark before the details: I'm aware of definitions for general $n$ which should, in theory, specialize to this $n=2$ case.  I'm mainly interested in definitions which are specific to $n=2$ are are close is spirit to this one.

Let $A$ be a weak 2-category.  ("Weak" meaning that composition of 1-morphisms is not strictly associative.)  I have in mind the case where the 2-morphisms of $A$ form vector spaces, but I don't think that matters for what follows.
Define a left representation of $A$ to be a (weak) 2-functor from $A$ to the 2-category of (1-categories, functors, natural transformations).  For each object $x$ of $A$ we have a 1-category $Y_x$.  For each 1-morphism $e:x\to y$ of $A$ we have a (notationally overloaded) functor $e: Y_y\to Y_x$.  For each 2-morphism $h:e\to f$ of $A$ we have natural transformation (also denoted $h$) between the functors assigned to $e$ and $f$.  For each pair of composable 1-morphisms $e$ and $f$ there are also an invertible natural transformation $c_{ef}$ which relates the functor for $ef$ to the composition of the $e$ functor and the $f$ functor.  All of this data is required to satisfy some standard relations.
Let $A^{op}$ denote the 2-category where the order of composition of 1-morphisms (but not 2-morphisms) is reversed.  (Reverse horizontally but not vertically.)
Let $\{W_{xy}\}$ be a collection of 1-categories with the structure of a left $A^{op}\times A$ representation.  For each object $(x,y)$ of $A^{op}\times A$ we have a 1-category $W_{xy}$.  For each pair of 1-morphisms $e:x\to u$ and $f:v\to y$ we have a functor $e\times f$ from $W_{xy}$ to $W_{uv}$.  And so on.
Now let's define the 2-coend of $\{W_{xy}\}$ as above.  The data for the 2-coend consist of (1) a 1-category $C$; (2) for each object $x$ of $A$ a functor
$$
 gl_x : W_{xx} \to C ;
$$
and (3) for each 1-morphism $e:x\to y$ of $A$ an invertible natural transformation $s_e$ between the functors $(e\times 1)\bullet gl_y$ and $(1\times e)\bullet gl_x$, as shown in the following diagram.

(source)
(My convention in the above sort of 2-dimensional commutative diagram is to enclose the natural transformations labeling the 2-cells in boxes.)
The above data is required to satisfy conditions 1 and 2 below, and also to be universal in the sense described below.
Condition 1:  For each 2-morphism $h:e\to f$ of $A$, the following 2-sphere-shaped diagram commutes.

(source)
(The natural transformation $s_f$ labels the "2-cell at infinity".  Saying that a diagram like this commutes means that two different natural transformations, built out of two subsets of $s_e$, $s_f$, $1\times h$, $h\times 1$ and some identity natural transformations, are equal.
Condition 2:  For each composable pair of 1-morphisms $e:x\to y$ and $f:y\to z$ of $A$, the following 2-sphere-shaped diagram commutes.

(source)
The data $(C, \{gl_x\}, \{s_e\})$ is required to be universal in the following sense.  For any $(C', \{gl'_x\}, \{s'_e\})$ satisfying conditions 1 and 2 above, there exists a functor $\theta: C\to C'$ and, for all objects $x$ of $A$, a natural transformation $\eta_x: \theta\circ gl_x\to gl'_x$, such that the following diagram commutes for all 1-morphisms $e$ of $A$.

(source)

Some further remarks.


*

*If we reverse all the arrows above, we get a definition of the 2-end of $\{W_{xy}\}$.

*The above abstract definitions in terms of a universal property can be converted into more concrete definitions in terms of generators and relations.  In this version the 2-coend $C$ has (by definition) objects the union over all $x$ of the objects of $W_{xx}$.  The morphisms of $C$ are the union of all morphisms of $\{W_{xx}\}$, plus addition invertible morphisms $s_{eu}: e\cdot u \to u\cdot e$, for all $e:x\to y$ and all objects $u$ of $W_{xy}$.  These extra morphisms must satisfy relations corresponding to conditions 1 and 2.  An object of the 2-end is a tuple $(r_x)$, indexed by objects $x$ of A, where each $r_x$ is an object of $W_{xx}$.  In addition, each such tuple is equiped with a collection of invertible morphisms $t_{er}: e\cdot r_y\to r_x\cdot e$ of $W_{xy}$.  A morphism between two such enhanced tuples is a tuple of morphisms which is compatible with the $\{t_{er}\}$ according to conditions 1 and 2.

*In the special case where $A$ is a tensor category and $W$ is the regular 2-category bimodule ${}_AA_A$, then the 2-end is the Drinfeld center of A and the 2-coend is what I would call the Drinfeld double of $A$.
 A: I don't think I have seen anything like this published before, but I have written up a similar definition here (see here too).
One thing in your definition I would take issue with is that your 2-coend's universal property needs more to deserve the '2-' in its name: what I would expect (and what my definition, which otherwise seems to be equivalent to yours, says) is a bicategorical universal property that says


*

*for $(C', \mathrm{gl'}, \ldots)$ as above, $\eta$ exists and is invertible, and

*for $\theta, \theta' \colon C \to C'$, any 2-cell (i.e. modification between extranaturals/wedges) $\theta \circ \mathrm{gl} \to \theta' \circ \mathrm{gl}$ is equal to $t \circ \mathrm{gl}$ for a unique $t \colon \theta \to \theta'$.

A: In addition to  Buschi Sergio's comment, Bozapalides' paper was one of the inputs into the paper by Jean-Marc Cordier and myself:
Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54.
In there, we generalised his construction to a homotopy coherent end /  coend. The construction as an indexed end /  coend has been used by various people (several of whom are likely to add references to this and I will not attempt to do that for them, :-) ) who have worked on homotopy ends and coends.  These formulations will not be optimised for your use but should give you the answers that you need. They are more like the infinity categorification of ends / coends, but the theory we developed may prove useful.
(I would be fascinated to find out how you hope to link them to TQFTs, as I have had thoughts in that direction, ..., so do keep in touch! I will look at the text that you linked to.)
(Edit: I checked and there is some discussion of simplicially enriched homotopy coherent ends and coends in the longer versions of the Crossed Menagerie, but not on the version linked from my nLab pages.  I can send you a copy if you wish.just contact me.)
A: Fosco Loregian has surveyed the theory of co/lax co/ends in Section 7.1 of his new book, citing Bozapalides's PhD thesis and paper on the subject (see also [MO67083]).
Another reference is Chapter 2 of the Alexander Corner's PhD thesis or [arXiv:1709.01332]. These two references develop the theory of extrapseudonatural transformations, defining bicoends (called "bicodescent objects" there) as universal extrapseudonatural transformations. Corner also cites the following papers:

[Street, 1976] Street, R. (1976). Limits indexed by category-valued 2-functors. J. Pure Appl. Al-gebra, 8(2):149–181.
[Street, 1980] Street, R. (1980). Fibrations in bicategories. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 21(2):111–160
[Lack, 2002] Lack, S. (2002). Codescent objects and coherence. J. Pure Appl. Algebra, 175(1-3):223–241. Special volume celebrating the 70th birthday of Professor Max Kelly.
[Street, 1987] Street, R. (1987). Correction to fibrations in bicategories. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 28(1):53–56

