Let $\bf F$ be differential 2-form on a 4-dimensional Lorentzian pseudo-Euclidean manifold $M$ with signature (3, 1) endowed with coordinate functions (t, x, y, z), where t increases in the dimension for which the metric is negative. Let $S$ be a cylindrical surface in M, whose circular faces are parallel to $dx \wedge dy$ and whose sides extend in the $t$ direction from $t=0$ to $t=1$.
Consider the integral of $\bf F$ over $S$. This integral involves the $dx \wedge dy$, $dt \wedge dx$, and $dt \wedge dy$ components of $\bf F$, and can be broken into 3 parts: the two disks, and the side of the cylinder.
Now I can compute the magnitude of each part easily, but I can't figure out the relative orientation. (On which side of the cylinder) does $dt \wedge dx$ face into the cylinder? (On which side) does $- dt \wedge dx$ face in? What about the two circular faces? Does $dx \wedge dy$ face into positive $t$ (and therefore the disk at $t=0$ faces into the cylinder, while the one at $t=1$ faces out)? Or does the circular face at $t=0$ face out of the cylinder when computing this integral?
What if the sign convention flips? And how does one generally determine the orientation of submanifold segments when computing closed integrals on $M$?
Note: I'm using the $-+++$ sign convention, but please specify how the answer changes when you switch sign conventions.
From what i understood, your setting is that of $M = (\mathbb{R}^{1,3}, \eta)$, Minkowski four-space together with its metric $\eta$, and want to consider the surface parametrized in the global coordinates $(t,x,y,z)$ as
$$S = D_1 \cup C \cup D_2 = \{(0,x,y,0)|x^2 +y^2 < 1\} \cup \{(t,x,y,0)|t \in [0,1], x^2+y^2 =1\} \cup \{(0,x,y,0)|x^2 +y^2 < 1\}$$
Furthermore assume there is a two-form $F$. It can be pulled back to $S$.
Then $$ \int_S F = \int_{D_1} F + \int_{C} F + \int_{D_2} F = \\ = \int_{x^2+y^2 < 1} F_{xy}(0,x,y,0) dx \wedge dy + \\ + \int_{0 \leq t \leq 1,x^2 + y^2 =1 } (F_{tx}(t,x,y,0) dt\wedge dx + F_{ty}(t,x,y,0) dt\wedge dy) + \\ + \int_{x^2+y^2 < 1} F_{xy}(1,x,y,0) dx \wedge dy $$ There are no choices here.
However, you may orient $S$, that is, choose $(\pm1)$ for each point in $S$ such that it varies smoothly. Since $D_1,C,D_2$ are all connected and orientable, you have three choices of sign. This depends on the physics you are considering. For example, if you want to compute the "magnetic charge" $\int F$ contained in the interior of $S$, you should take $ -D_1 + C + D_2$, but you may have different things in mind.
All of this is independent of the signature of the metric - the metric was not needed.