No,
question 1: No, take any finite field extension of $\mathbb{Q}$. It is basic but has a simple module that is not 1-dimensional and thus it is not of the form $KQ/I$ for $I$ admissible (since all simple modules are 1-dimensional for algebras of the form $KQ/I$).

For any field $K$, a basic algebra is isomorphic to a quiver algebra with admissible ideal if and only if the algebra $A$ is split (also called elementary sometimes), meaning that $A/J$ is isomorphic to matrix algebras over $K$.


Question 2: The best (computer) tool is the GAP-package qpa: https://folk.ntnu.no/oyvinso/QPA/ . Given a quiver and admissible ideal, you can calculate many information for the algebra, including the dimension.