If we add a constant weight vector with an absolute function, does it still remain convex then? We know that absolute functions are convex. Now what if we add a constant weight vector to it, does it still remain convex?
Say the equation is Absolute loss regression + L1 regularization, we know that absolute loss regression is a convex function, so does adding L1 regularization preserves its convexity?
 A: A function is convex if it obeys the following inequality, for $\{0<a<1,  a+b=1\}$:
$$f(ax+by)\leq af(x)+bf(y)$$
Restricting the inequality we can define strict convex functions as the ones that obey
$$f(ax+by)\lt af(x)+bf(y)$$
If $f(x) = \sqrt{x^2}=|x|$, then by the triangle inequality:
$$f(ax+by)=|ax+by|\leq a|x|+b|y|$$
Thus the absolute value is a convex function (though not strictly convex), as you correctly pointed out.
The addition of convex functions is convex, so a objective like $\mathcal L = |\beta^TX-Y|+\lambda|\beta|$ is a convex function of $\beta$.
Now, onto convex optimization.
In convex optimization, all local minimums are global minima. $\mathcal L$ is convex, so it has a single minimum as well: the global minimum.  Thus it admits an unique optimum value. It does not guarantee uniqueness of the solution $\hat \beta$ however, only strictly convex functions admit a single solution. This means that it is possible for different $\hat \beta$ to achieve the same optimum.
So, the answer is yes, it remains convex.
