Let's assume that $X$ and $Y$ are two independent random variables with density. Let $c\in\mathbb R$ be some constant.
Question: Is it true that the conditional law of $X+Y|X=c$ is the law of $c+Y$ ?
For me, it seems intuitively clear but trying to compute the conditional density of $X+Y|X$ leads me to the joint density of $(X+Y,X)$ where I'm stuck.
Yes it's true. We can write the joint density of $X$ and $Y$ as $$ f_{X,Y}(x,y) = f_X(x)f_Y(y).$$
We can do a change of variables $$ U = X, V = X+Y$$ to get the joint distribution of $X$ and $X+Y.$ So $X = U$ and $Y = V-U,$ and the Jacobian factor is $1.$ We have $$f_{U,V}(u,v) = |J| f_{X,Y}(u,v-u) = f_X(u)f_Y(v-u). $$ The marginal PDF of $U$ is of course $f_X,$ so we have $$ f_{V\mid U}(v\mid u) = \frac{f_X(u)f_Y(v-u)}{f_X(u)} = f_Y(v-u).$$
Similarly the PDF of $u+Y$ (where $u$ is a constant), is just the PDF of $Y$ shifted to the right by $u,$ i.e. $f_Y(y-u).$
If $X$ and $Y$ weren't independent we would have that the conditional law $X+Y\mid X = c$ is the conditonal law of $c+Y\mid X=c.$