Q1. Yes, such 3 monomials exist. Moreover, the generic homogeneous quadratic polynomials $f,g,h$ are such that the dimension of the quotient algebra $K⟨x,y,z⟩/(f,g,h)$ is 28. This follows, in particular, from Th.1.3 of [Natalia Iyudu and Stanislav Shkarin. The Golod-Shafarevich inequality for Hilbert series of
quadratic algebras and the Anick conjecture].

Q2. In contrast, such an ideal does not exist if $n\ge 5$. If follows from the Vinberg version of the Golod--Shafarevich theorem that if an ideal $I$ is generated by $r$ elements (which are linear combinations of at least quadratic monomials) in a free associative algebra $F$ with $g$ generators, then the quotient algebra is infinite-dimensional provided that the polynomial $1-gz+rz^2$ has a positive root (cf, for example, Prop. 2.7 in [M.Ershov, Golod-Shafarevich groups: a survey]). In your case we have $r=g=n-1$, so that the quotient algebra is infinite if $n\ge 5$.