a question about the Collatz conjecture (relation of smallest number in a cycle and minimal cycle-length) I've only done a bit of research on the current findings, not sure if anyone here can answer this.              

Q1: I just haven't been able to find,     


  
  has it been shown yet that a it is impossible for a loop to exist,      
  and for a counter example to exist does it have to grow to infinity?         
  


Q2: I've seen that all numbers up to 2^60 have been checked, and using this information I'm able to show that no loops exist of length 101 or less (as a lower bound).   


  Has something like this been shown before? I just can't seem to find much information on it.
It has neither been shown that a loop other than the trivial $\{4,2,1\}$ does not exist, nor that there exists a trajectory that grows to infinity.
Reformulate the Collatz iteration in a compact form as

(1) $v_n = z_n + 2^{2n+1-r}m_r$

for any input $u \in 2\mathbb{N}+1 \not \equiv 0\pmod{3}$ such that $u=3m_r+r$ where $r=1$ if $u \equiv 1\pmod{3}$ or $r=2$ if $u \equiv 2\pmod{3}$ and $z_n = \sum_{i=1}^{n} 2^{i-1}$ for $n \in \mathbb{N}$. Iterating (1) by one step for each $n \in \mathbb{N}$ yields an infinite set $\{v_n\}_{n \in \mathbb{N}}$, termed sibling set, arising from parent $u$. Suppose we consider $u$ and $v_n$ as vertices connected by edge $(u, v_n)$ from $u$ to $v_n$ for every $n \in \mathbb{N}$. Then starting with $u_0=1$, repeatedly iterating (1) constructs an arborescence graph $G=(V,E)$ that is the union of infinitely many sibling sets, with root at $r_0$, vertex set $V$ and edge set $E$, as shown here. Every path of any length from the root in $G$ represents an odd Collatz sequence in reverse. It can be shown that every vertex in $G$ is unique and that $V=2\mathbb{N}+1$, i.e. $V$ exactly covers the set of odd natural numbers. Since $G$ is weakly connected, then we infer that: (i) only the trivial loop exists (but is necessarily ignored in $G$) and (ii) every odd Collatz sequence always converges to $1$ (hence, cannot diverge). These results, subject to further scrutiny, answer both questions under Q1 above.