Given the Ito SDE $$ dX_t=a(X_t,t)dt + b(X_t,t) dB_t $$ where $a(X_t,t)$ and $ b(X_t,t)$ satisfy the Lipschitz condition for existence and uniqueness of solutions. Given a function $f(X_t,t) ∈ C^2$ using Ito's formula I can derive the SDE
$$ df = \frac{\partial f} {\partial t} dt + \frac{\partial f}{\partial x} dX_t + \frac{1}{2} \frac{\partial^2 f}{\partial x^2} dX_t^2 $$
where $dX_t^2$ is computed using Ito's lemma.
The question is: are there any requirements that $f(X_t,t)$ must satisfy to guaranty the existence and uniqueness of solutions (I would say yes).
Any reference is welcome. Thanks in advance.
You shall distinguish between the equation and the 'direct' definition of something. For example, whenever you have an expression of the form $$ x = g(x) \tag{1} $$ where $g$ is a certain function/operator that is given to you, you may be asked to find $x$ which satisfies such expression. You can never be sure whether such $x$ exists, or whether there is only one such $x$. Indeed, if you change the $x$ in the RHS, the LHS changes as well since it depends on $x$. Anyway, suppose we found such $x$ and it is unique.
Now, imagine that you also have an expression $$ y = h(x) \tag{2}. $$ This is not an equation, but rather a definition of $y$. Indeed, to find the value of $y$ the only thing we need to do is to substitute $x$ (found on the previous step) as an argument of $h$ - that's it.
What do you have in your original post (OP) is that $x$ is the process $(X_t)_{t\geq 0}$ which satisfies $$ X_T = X_0 + \int_0^Ta(X_t,t)\mathrm dt + \int_0^Tb(X_t,t)\mathrm dB_t \tag{$1^\prime$} $$ which can be compactly written through differentials as in your case. Here the operator $g$ as in $(1)$ is just these integrals and functions $a,b$ applied to $X$. Again, the LHS and the RHS both contain $X$, so that it's not obvious whether there exists such $X$ which makes both sides be equal. Thus, we need conditions on $a$ and $b$ to assure such existence, or uniqueness.
Now, let $Y_t = f(X_t,t)$ be another process. For $f\in C^2$ we can use an alternative definition of $Y$: $$ Y_T = Y_0 + \int_0^T\frac{\partial f}{\partial t}(X_t,t)\mathrm dt + \int_0^T \frac{\partial f}{\partial X}(X_t,t)\mathrm dX_t + \int_0^T \frac{\partial^2 f}{\partial X^2}(X_t,t)\mathrm dX_t^2 \tag{$2^\prime$} $$ which yet again can be written in a compact symbolic form as via differentials as you did. Here you can think of the RHS as a function of the process $X$: $h((X_t)_{t\geq 0})$. It is only used to define $Y$. The only thing that you need to take care of is that $h$ is defined for a particular value of the argument, that is all the integrals in the RHS of $(2')$ are well-defined. Actually, here the only Ito integral is the middle one (a part of $\mathrm dX_t$) and you indeed shall check that $$ Z_t:=b(X_t,t)\frac{\partial f}{\partial X}(X_t,t) $$ satisfies for example $(iii)$ in Definition 3.1.4 or $(iii)'$ in Definition 3.3.2 of Oksendal: "Stochastic Differential Equations".
As a result, you actually shall not talk about solutions of $(2')$ as much as you don't talk about the solutions of $y = 3$. Instead, you talk about solutions of $x = x^2+1$.