Let $p_1, ..., p_m \colon \mathbb{R}^n \to \mathbb{R}$ be $m < n$ linearly independent homogeneous polynomials each of degree exactly $d$, and assume they have nonzero intersection. Is the codimension of the affine variety determined by $p_1, ..., p_m$ at least $m$?
If this does not hold in general, can we lower bound the codimension of the variety in terms of $d$ (e.g., what if $d=2$)?
No, this is not true, in any form. First, over $\mathbb{R}$, every algebraic variety is defined by a single equation, regardless of its co/dimension. The reason is because over $\mathbb{R}$, polynomials $p_1, ..., p_m$ vanish if and only if $p_1^2 + ... + p_m^2$ vanishes!
But the situation is not better over $\mathbb{C}$ either, now there are simply new issues. For example, there is the twisted cubic curve, which is a curve in $\mathbb{CP}^3$ which is cut out by exactly three equations, all of which are quadratic equations, and no two of them suffice, so the codimension is $2$, rather than the expected $3$. In general, varieties with the property you ask for are called "complete intersections," and are related to a slew of algebraic properties of the ideal they generate inside $\mathbb{C}[x_1, ..., x_n]$, or its homogenous counterpart. For example, if the polynomials generate a regular sequence, then you have at least a local complete intersection.
Unfortunately, I don't know any super short exposition of these ideas. There are a bunch of moving parts, some algebraic and some geometric, and I learned them from Eisenbud's Commutative Algebra, and Hartshorne's Algebraic Geometry.