Let $\Omega$ be a bounded open set in $\mathbb{R}^n$ with $C^1$ boundary. I know the following (possible) definition for $$\int_{\partial\Omega}u\,d\sigma.$$ For each $x_0\in\partial\Omega$, there exist an open subset $U$ in $\mathbb{R}^n$ containing $x_0$, an open subset $A$ in $\mathbb{R}^{n-1}$ and a $C^1$ function $g:A\rightarrow\mathbb{R}$ such that its graph is $G_g=\partial\Omega\cap U$. By compactness of $\partial\Omega$, we can write $\partial\Omega=\cup_{j=1}^N U_j$, where each $U_j$ is open in $\mathbb{R}^n$, and $\partial\Omega\cap U_j=G_{g_j}$, where $g_j:A_j\rightarrow\mathbb{R}$ is $C^1$. Take a partition of unity $\{\eta_j\}_{j=1}^N$ corresponding to $\{U_j\}_{j=1}^N$: $\eta_j\in C_c^{\infty}(U_j)$, $0\leq\eta_j\leq 1$ and $\sum_{j=1}^N\eta_j=1$ on $\partial \Omega$.
Let $u:\partial\Omega\rightarrow\mathbb{R}$ measurable. Define $u_j=u\,\eta_j$, which has support in $U_j$. Then: $$\int_{\partial\Omega}u\,d\sigma=\sum_{j=1}^N\int_{A_j}u_j(x',g_j(x'))\sqrt{1+|\nabla g_j(x')|^2}\,dx'$$(whenever the right-hand side exists).
My two doubts are the following:
Given a subset $\Gamma$ of $\partial\Omega$, in what sense it is understood $\int_\Gamma u\,d\sigma$? Is it true that $$\int_{\partial\Omega}u\,1_\Gamma\,d\sigma=\int_\Gamma u\,d\sigma\,?$$ ($1_\Gamma$ is the characteristic function on $\Gamma$).
If $u_m\rightarrow u$ in $L^p(\partial\Omega)$, is it true that there exists a subsequence $u_{m_k}$ that converges pointwise almost everywhere on $\partial\Omega$?
The answer to your questions is yes. Using the area formula for Lipschitz functions (you can find it in the book of Evans and Gariepy Evans and Gariepy ) you can show that $$\int_{\partial\Omega}u\,d\sigma=\int_{\partial\Omega}u\,d\mathcal{H^{n-1}},$$ where $\mathcal{H^{n-1}}$ is the $n-1$ dimensional Hausdorff measure. Since $\mathcal{H^{n-1}}$ is a measure, all standard properties of integration and $L^p$ spaces hold for the surface integral.