Gauge equivalence of Lie-valued forms on the base space of a principal bundle Given a principal $G$-bundle $P\xrightarrow{\pi} M$:
Assuming the bundle is globally trivial, we define two Lie$G$-valued 1-forms $A_1,A_2$ on $M$ to be gauge-equivalent
if there is a principal bundle connection $\tilde{A}$ on $P$
and two global sections $s_j$ of $\pi$ such that $s_j^* \tilde{A} = A_j$, for $j=1,2$.
(The above seems to reflect the definitions used in physics, if I'm not mistaken.)


*Is the above the correct definition of gauge equivalence?


*Is every Lie$G$-valued 1-form on $M$ equal to $s^*\tilde{A}$ for some global section $s$ and connection $\tilde{A}$ on $P$?


*What would be the definition of gauge-equivalence when the bundle isn't trivial?
 A: I question that there is $the~correct$ definition, but indeed, your definition is equivalent to the usual one if the setting is restricted to trivial bundles.
So in general, a connection form $\mathcal{A}$ is a $\mathfrak{g}=\text{Lie}(G)$-valued differential form on a principal $G$-bungle $\pi:P\to M$ with some properties. Moreover, for each local section $s:U\to P$ with $U\subset M$ you can define the pullback $s^*\mathcal{A}$ to obtain a $\mathfrak{g}$-valued 1-form on $U$. If you have have a family of section $s_i:U_i\to P$ such that $\bigcup_i U_i=M$, you accordingly obtain a family $\{\mathcal{A}_i=s_i^*\mathcal{A}\}_i$. You will notice that under "chart transitions" the local connection forms $\{\mathcal{A}_i\}$ transform as
$$
\mathcal{A}_j=\text{Ad}_{g_{ij}^{-1}(\cdot)}\circ\mathcal{A}_i+(g_{ij})^*\mu^\text{MC}
$$
where $g_{ij}:U_i\cap U_j\to G$ is the transition function between $s_i$ and $s_j$ (i.e. the map $U_i\cap U_j\to G$ such that $s_j(x)=s_i(x)g_{ij}(x)$ holds) and $\mu^\text{MC}$ is the Maurer-Cartan form of $G$.
Conversely, for a given atlas $\{(U_i,\phi_i)\}_i$ of the principal bundle and a family of $\mathfrak{g}$-valued 1-forms $\{\mathcal{A}_i\}_i$ with $\mathcal{A}_i\in\Omega^1(U_i;\mathfrak{g})$, you can show that if the local forms $\{\mathcal{A}_i\}$ obey the above transformation rule (with a certain choice of the $g_{ij}$), then you can patch these local forms together to obtain a global connection 1-form $\mathcal{A}\in\Omega^1(P;\mathfrak{g})$. This procedure is inverse to the above, that is, patching together some $\{s_i^*\mathcal{A}\}$ (which already obey the transformation law) results in $\mathcal{A}$ again. So you have a local description and a global description of a connection form.
Finally, in the global setting, two connection 1-forms $\mathcal{A},\widetilde{\mathcal{A}}\in\Omega^1(P;\mathfrak{g})$ are said to be gauge equivalent if there exists a gauge transformation $\mathfrak{f}:P\to P$ (a.k.a. a $G$-bundle isomorphism a.k.a. chart transition) such that $\mathfrak{f}^*\mathcal{A}=\widetilde{\mathcal{A}}$ holds.
So to you concrete questions:
To (1.): Indeed, if you have a $\mathfrak{g}$-valued 1-form $\mathcal{A}$ on $M$, you can define a $\mathfrak{g}$-valued 1-form $\widetilde{\mathcal{A}}$ on the trivial bundle $P=M\times G$ by the trivial choice. This corresponds to the above $patching~together$ (in the identity chart $P=M\times G\to M\times G$), wherefore I note that the transition rule for only one chart is an empty condition. Denote $s_e:M\to M\times G,~x\mapsto (x,e)$ with the identity element $e\in G$. Then the afforementioned $trivial~choice$ is the unique choice such that $\mathcal{A}=s_e^*\widetilde{\mathcal{A}}$. But for any other section $s:M\to P$ you also obtain another connection 1-form $\widetilde{\widetilde{\mathcal{A}}}$ such that $\mathcal{A}=s^*\widetilde{\widetilde{\mathcal{A}}}$.
to (0.): What you define to be "gauge equivalent" for the forms $\mathcal{A}_1,\mathcal{A}_2\in\Omega^1(M;\mathfrak{g})$ is to claim that they are two local representation w.r.t. different sections $s_1,s_2$ of one and the same connection form $\widetilde{A}\in\Omega^1(P;\mathfrak{g})$, i.e. $\mathcal{A}_1=s_1^*\widetilde{\mathcal{A}}$ and $\mathcal{A}_2=s_2^*\widetilde{\mathcal{A}}$.
Constructing from the two sections in the former version a transition map $g:M\to G$ (as the $g_{ij}$ above) and the corresponding $G$-bundle isomorphism (a.k.a. gauge transformation) $\mathfrak{f}:P\to P$, you will notice that also $\mathcal{A}_2=s_1^*(\mathfrak{f}^*\widetilde{\mathcal{A}})$ holds, that is, $\mathcal{A}_1$ and $\mathcal{A}_2$ are local representations of two connection forms w.r.t. one and the same section $s_1$. More general, you  will never be able to tell whether two local representation belong to two connection forms in one section or to one connection form in two sections. That is one flaw of restricting to the local picture. However, this way you can see how your definition is, in the trivial bundle setting, equivalent with the usual definition above.
(2.) should be answered above in the global setting. For sure this can be broken down to the local language for the local representations in $\Omega^1(U;\mathfrak{g})$, but keeping the above inability to distinguish whether some $\mathfrak{f}:P\to P$ is viewed as chart transition or as gauge transformation, such a definition on the local level appears to me as somewhat useless.
