It is required to write a derivation to prove $\{(\phi\to\psi),(\phi\to\chi)\}\vdash (\phi\to(\psi\wedge\chi))$. The following is my attempt. Here, $\to E$ is Implication Elimination, $\wedge I$ is And Introduction, and $\to I$ is Implication Introduction.
$\underline{\underline{\cancel{\phi} \ \ \ \ (\phi\to\psi)}_{(\to E)} \ \ \ \ \underline{\require{cancel} \cancel\phi \ \ \ \ (\phi\to\chi)}_{(\to E)}\\ \ \ \ \ \ \ \ \ \ \psi \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \chi}_{(\wedge I)} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{(\psi\wedge\chi)}_{(\to I)} \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (\phi\to(\psi\wedge\chi)) $
Is the above derivation correct, and are there any rules in general to decide how to "group" the statements when writing a derivation?
The natural deduction derivation is correct. You assume $\phi$ to eliminate the conditional from both premises, introduce a conjunction on the results, and then discharge that assumption by introducing a conditional again.
Various notations are used for calculus trees, but most will mark which assumptions are discharged and indicate what rule does so; for example, by using brackets and an indices, like so:$$\dfrac{\dfrac{\dfrac{[\phi]^1~~(\phi\to\psi)}{\psi}{\small\to\!\!\sf E}~~\dfrac{[\phi]^1~~(\phi\to\chi)}{\chi}{\small\to\!\!\sf E}}{\psi\land\chi}{\small\land\sf I}}{\phi\to(\psi\land\chi)}{\small\to\!\!\sf I^1}$$
Crossing off the discharged assumptions also works, but it is imperative to indicate what rule does the discharge of wich assumption. It helps readers decipher the logic, particularly in derivations with multiple and nested discharges.
In a sequent calculus tree, active assumptions are tracked on each line, which is a bit cluttered:
$$\dfrac{\dfrac{\dfrac{\dfrac{}{\phi\to\psi\vdash\phi\to\psi}{\small\sf Id}}{\phi,\phi\to\psi\vdash\psi}{\small\sf\to\!\!E}\quad\dfrac{\dfrac{}{\phi\to\chi\vdash\phi\to\chi}{\small\sf Id}}{\phi,\phi\to\chi\vdash\chi}{\small\sf\to\!\!E}}{\phi,\phi\to\psi,\phi\to\chi\vdash \psi\land\chi}{\small\sf\land I}}{\phi\to\psi,\phi\to\chi\vdash \phi\to(\psi\land\chi)}{\small\sf\to\!\!I}$$
Fitch style notations use nested boxes (or paragraph indentations) to track the rise and discharge of assumptions, and use line numbering to reference which statements are used for inferences. $$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline #2\end{array}}\fitch{~~1.~~\phi\to\psi\\~~2.~~\phi\to\chi}{\fitch{~~3.~~\phi}{~~4.~~\psi\qquad\to\mathsf E(3,1)\\~~5.~~\chi\qquad\to\mathsf E(3,2)\\~~6.~~\psi\land\chi\qquad\land\mathsf I(4,5)}\\~~7.~~\phi\to(\psi\land\chi)\qquad\to\mathsf I(3-6)}$$