First inequality:
Expand to obtain the inequality $f(a,b,c,d):=a^2b^2c^2+b^2c^2d^2+c^2d^2a^2+d^2a^2b^2 \leq 1$; we will prove this for any nonnegative reals $a,b,c,d$ with $a+b+c+d=3$. Since the set $\{(a,b,c,d) \in \mathbb{R}^4 \mid a+b+c+d=3\} \cap [0,3]^4$ is compact, $f(a,b,c,d)$ attains a global maximum over it. I claim that, when $f$ attains this maximum, any two values are either equal, or one of them is $0$.
Suppose that we fix $c$ and $d$ and let $a,b$ vary with $a+b:=t$ fixed. I claim that $f(a,b,c,d)$ is maximized either $a=b=t/2$ or one of $a,b$ is $0$ (so the other equals $t$), and that this maximization is "strict", i.e. $\max\{f(t/2,t/2,c,d),f(t,0,c,d)\}>f(a',b',c,d)$ for any $\{a',b'\} \neq \{t,0\},\{t/2,t/2\}$. Rewrite $f(a,b,c,d)$ as
$$(ab)^2(c^2+d^2)+(a^2+b^2)c^2d^2=(ab)^2(c^2+d^2)+(t^2-2ab)c^2d^2,$$so since $t,c,d$ are all fixed we want to investigate the maximum of $(c^2+d^2)(ab)^2-2cd(ab)$. This is a quadratic in $ab$ with positive leading coefficient, so on any interval it is maximized at an endpoint, i.e. we either want to minimize $ab$, which clearly occurs when one of $a,b$ is $0$, or maximize it, which clearly occurs when $a=b$.
Therefore, if $f(a,b,c,d)$ is a unique maximum, and some two variables, WLOG $a$ and $b$, are different and both nonzero, then we can apply the above claim to find a larger value of $f(a,b,c,d)$: contradiction. Therefore, we only have to prove the inequality when some of the variables are zero, and the other values are all equal.
If two or more variables equal $0$, then it is clear that $f(a,b,c,d)=0$ as well. If one variable equals zero, WLOG $d$, then $f(a,b,c,d)=a^2b^2c^2=1$. If no variables equal zero, then $a=b=c=d=\tfrac{3}{4}$, and $f(a,b,c,d)=\tfrac{3^6}{4^5}=\tfrac{729}{1024}<1$. Therefore, $f(a,b,c,d) \leq 1$ for all nonnegative reals $a,b,c,d$ with $a+b+c+d=3$, as desired. $\blacksquare$
The second inequality is similar:
Define $g(a,b,c,d)=\sum_{\mathrm{cyc}} a^3b^3c^3$, which we want to prove is at most $1$. Again, $g(a,b,c,d)$ attains a unique maximum, and I again claim that when this happens any two variables are either equal or $0$. Fix $c,d$ and let $a,b$ vary with $a+b:=t$ fixed. Then
$$g(a,b,c,d)=(ab)^3(c^3+d^3)+(a^3+b^3)c^3d^3=(ab)^3(c^3+d^3)-(t^3-3t(ab))(c^3d^3)=(c^3+d^3)(ab)^3-3tc^3d^3(ab)+\text{constant},$$and as a cubic in $ab$ this will be "strictly" maximized at an endpoint on any interval which is a subset of $\mathbb{R}_{\geq 0}$ (and $ab \geq 0$, so this is fine).
Finally, we can check the same three cases as before to get that $g(a,b,c,d) \leq 1$ for all nonnegative reals $a,b,c,d$ with $a+b+c+d=3$, as desired. $\blacksquare$