I stumbled upon a definition of a heavy tail (on the left side) as follows

$\lim_{x\to\infty}( [1-F(x)]e^{\lambda x})=\infty$

for $\lambda>0$. Now on-to questions:

1. what does it mean if I get $-\infty$? Is it still not heavy-tailed? What if I get zero?

2. Is this statement equivalent to - only these distributions have infinite moments?

3. Is this definition of heavy-tail correct for left side tail?

$\lim_{x\to-\infty}( F(x)e^{\lambda x})=\infty$

1. what does it mean if I get $-\infty$? Is it still not heavy-tailed? What if I get zero?

You cannot get $-\infty$ since cumulative distribution function is a probability, and it cannot increase $1$.

1. Is this statement equivalent to - only these distributions have infinite moments?

It is equivalent to: only this distributions have infinite exponential moments $\mathbb Ee^{\lambda X}=\infty$ for any $\lambda >0$.

Say, Pareto distribution with probability density function $f(x)=\frac{5}{x^6}$ for $x\geq 1$ has four finite moments $\mathbb EX^4<\infty$ but higher moments are infinite.

The distribution with CDF $F_X(x)=1-e^{-\sqrt{x}}$, $x\geq 0$ has all power moments finite: $\mathbb EX^k<\infty$ for any natural $k$, but $$\lim_{x\to\infty}( [1-F(x)]e^{\lambda x})=\lim_{x\to\infty}e^{-\sqrt{x}}e^{\lambda x}=\infty$$ for any $\lambda >0$.

1. Is this definition of heavy-tail correct for left side tail? $$\lim_{x\to-\infty}( F(x)e^{\lambda x})=\infty$$

If you mean "for any $\lambda <0$" then it is correct and is equivalent to non-existance of exponential moments $\mathbb E e^{\lambda X}$ for $\lambda<0$. Note that for $\lambda>0$ the condition $$\lim_{x\to-\infty}( F(x)e^{\lambda x})=\infty$$ is impossible: both multipliers tend to zero while $x\to-\infty$.