What progress has been made on the Collatz conjecture since Crandall's 1978 paper?

Question

I have recently read the famous paper by Crandall (1978) on the $3x+1$ problem, and I wonder what progress has been made since then.

The paper claims that:

1. If a cycle exists, then the minimum number of $3x+1$ steps for the number to reach itself is $17985$. Is there a better lower bound nowadays?
2. It is not known whether a positive density of odd integers satisfy the conjecture. Is that still unknown?