Does every even-dimensional Lie group admit a complex structure?

This answers the question only tangentially; namely, I'll address whether every Lie group has an invariant complex structure. The answer to this question is no. Let's recall the standard construction. Say we have a real Lie group $G$ of dimension $2n$ and a complex structure on its Lie algebra $\mathfrak{g} = \mathrm{Lie}(G) \cong T_1G$, that is, an $\mathbb{R}$-linear endomorphism $J_1\colon T_1G\to T_1G$ such that $J_1^2 = -\mathrm{id}_{T_1G}$. We want to give $G$ the structure of an almost complex manifold that restricts to $J_1$ on $T_1G$ and such that the left-multiplication, $\ell_g\colon G\to G$, $h\mapsto gh$, has $\mathbb{C}$-linear differential (is almost-holomorphic?) for each $g\in G$. It's easy to see that there is only one such almost complex structure $J\colon TG\to TG$; more concretely, over $g\in G$ it is given by the map $J_g := (\ell_{g})_*\circ J_1\circ (\ell_g^{-1})_*\colon T_gG\to T_gG$. In particular, if $X\colon G\to TG$ is the left-invariant vector field associated with $v\in T_1G$, i.e., $X_g = (\ell_g)_*(v)$, then $(JX)_g = (l_g)_*\circ J_1(v)$ and so $JX$ is just the left-invariant vector field associated with $J_1(v)$.

The question now is: when is this almost complex structure integrable?

The Newlander–Nirenberg theorem states that an almost complex structure $I\colon TG\to TG$ is integrable if and only if its Nijenhuis tensor $$[X,Y]+I[IX,Y]+I[X,IY]-[IX,IY]$$ vanishes for all vector fields $X,Y$. Since the Nijenhuis tensor only depends on the point-wise values of the vector fields, it is seen to be trivial as soon as it vanishes on all left-invariant vector fields. Thus, by construction:

Proposition— Let $J$ be a complex structure on $\mathfrak{g}$. Then the left-invariant almost complex structure on $G$ induced from $J$ is integrable if and only if $[JX,JY] = [X,Y]+J[JX,Y]+J[X,JY]$ for all $X,Y\in\mathfrak{g}$.

We see that there is an obstruction to the integrability of an invariant almost complex structure, but this obstruction is seen entirely on the level of Lie algebras. Since every finite-dimensional real Lie algebra is the Lie algebra of some simply connected Lie group, it suffices to construct a Lie algebra which fails to admit a $J$ such that $J^2 = -\mathrm{id}_{\mathfrak{g}}$ and $[JX,JY] = [X,Y]+J[JX,Y]+J[X,JY]$ for all $X,Y\in\mathfrak{g}$.

Examples can be found, e.g., in work of Goze and Remm, Non existence of complex structures on filiform Lie algebras, arXiv:math/0103035v2. In fact, by Proposition 4, the following Lie algebras provide examples:

Given $n\geq 2$ let $\mathfrak{l}_{2n}$ be the $2n$-dimensional Lie algebra generated by $x_1,x_2,\dots ,x_{2n}$ and Lie bracket such that $[x_1,x_i] = x_{i+1}$ for all $i=2,3,\dots,2n-1$ and $[x_i,x_j] = 0$ for all other $i,j$. The strategy here is to show that if there were such an $J$ for $\mathfrak{l}_{2n}$, then $J(x_{2n})$ had to be central; but the center is generated by $x_{2n}$, hence $J(x_{2n}) = ax_{2n}$. But then $a^2 = -1$, a contradiction.

In the case of compact Lie groups, it is true that an even-dimensional Lie group always admits a complex structure. See Example 4.32 in Félix, Oprea, Tanré, Algebraic Models in Geometry (available here https://www.maths.ed.ac.uk/~v1ranick/papers/tanre.pdf).