Suppose that $M$ is an integer, symmetric matrix of order $n>2$ with the positive integers $K_1,\dotsc,K_n$ on its main diagonal, and with all the off-diagonal elements equal to $0$ or $1$ so that the number of the off-diagonal nonzero elements in row $i$ (therefore, also in column $i$) is $2K_i$, for each $i\in[1,n]$.
Q1: Given that all numbers $K_i$ are roughly about $n^{1/4}$, with $\max_i K_i<2\min_i K_i$, how small can the rank of $M$ be?
Q2: Under the same assumptions, how small can the rank be if $M$ is regarded a matrix over the finite field $\mathbb F_p$ with $p\approx n^{1/4}$?
Q3: Assuming, in addition, that $K_1=\dotsb=K_n$, can $M$ be singular?
For Q1, I can prove something like $\mathrm{rk}\, M>(c+o(1))n$, where $c=\min K_i/\max K_i$.
