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EDIT: It occurs to me that I didn't directly talk about twisted objects. The idea is the same. Let us start with an object $Y'$ over $U_0$ equipped with an isomorphism $f_0:Y'\rightarrow Y\times_XU_0$. This isomorphism has to satisfy the co-cycle condition, which means the following:

We have two projections $p_1,p_2:U_0\times_XU_0\rightarrow U_0$. So we have two different ways to pull-back $Y'$ to over $U_0\times_XU_0$, giving us $p_1^*Y'$ and $p_2^*Y'$. The two pull-backs of $Y\times_XU_0$ under these projections are canonically isomorphic, since the two projections $p_1,p_2$, when composed with the map $U_0\rightarrow X$ agree with the structure map for $U_0\times_XU_0\rightarrow X$. So pulling back $f_0$ gives us isomorphisms $$p_1^*Y'\rightarrow Y\times_X(U_0\times_XU_0)\rightarrow p_2^*Y'.$$

This is your $\alpha_{0,0}$; if it satisfies the co-cycle condition--this amounts to the required compatibility between the pull-backs of $\alpha_{0,0}$ to $U_0\times_{X}U_0\times_{X}U_0$ under the three different projections to $U_0\times_XU_0$--then it gives you descent data for $Y'$. In the Galois setting, if you use $f_0$ to identify $Y'$ with $Y\times_XU_0$, then $\alpha_{0,0}$ is giving you a twisted' action of $G$ on $Y\times_XU_0$.

This data is not always effective. I would highly recommend the chapter on descent in Bosch-Lutkebohmmert-Raynaud's Neron Models' for an explanation of all these things.

1

For your question 2, note that the fiber product $U_0\times_XU_0$ can be non-trivial (unlike the case of an open sub-set $U_0\subset X$, where it would be $U_0$ again) with two different projections $p_1$ and $p_2$ onto $U_0$ giving two different structures to it as a $U_0$-scheme. The isomorphism $\alpha_{0,0}$ is an isomorphism between these two different $U_0$-structures on $Y\times_XU_0\times_XU_0$ and is also a non-trivial piece of information

As an example, if $X=Spec\;k$ and $U_0=Spec\;K$ where $K/k$ is a Galois extension with Galois group $G$, then we get an isomorphism $G\times U_0\rightarrow U_0\times_XU_0$ using the group action. The two structures of $G\times U_0$ as a $U_0$-scheme correspond to the two maps $(g,u)\mapsto u$ and $(g,u)\mapsto gu$. The isomorphism $\alpha_{0,0}$ satisfying the co-cycle condition now equates precisely to giving an action of $G$ on $Y'$ compatible with its action on $U_0$. This is Galois descent. See Serre's Local fields, for example.