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For a positive integer $t\geq 2$, let $b_{t}(n)$ denote the number of $t$-regular partitions of a nonnegative integer $n$. Motivated by some recent conjectures of Keith and Zanello, we establish infinite families of congruences modulo $2$ for $b_9(n)$ and $b_{19}(n)$. We prove some specific cases of two conjectures of Keith and Zanello on self-similarities of $b_9(n)$ and $b_{19}(n)$ modulo $2$. We also relate $b_{t}(n)$ to the ordinary partition function, and prove that $b_{t}(n)$ satisfies the Ramanujan's famous congruences for some infinite families of $t$. For $t\in \{6,10,14,15,18,20,22,26,27,28\}$, Keith and Zanello conjectured that there are no integers $A>0$ and $B\geq 0$ for which $b_t(An+ B)\equiv 0\pmod 2$ for all $n\geq 0$. We prove that, for any $t\geq 2$ and prime $\ell$, there are infinitely many arithmetic progressions $An+B$ for which $\sum_{n=0}^{\infty}b_t(An+B)q^n\not\equiv0 \pmod{\ell}$. Next, we obtain quantitative estimates for the distributions of $b_{6}(n), b_{10}(n)$ and $b_{14}(n)$ modulo 2. We further study the odd densities of certain infinite families of eta-quotients related to the 7-regular and $13$-regular partition functions.