How many hex digits do I need to compare when manually checking hash functions?
How many hex digits do I need to compare when manually checking hash
  functions?


If you actually want the full security guarantees of the hash function to apply: all of them.


  I usually just look at the first/last 5 or 6 hex digits and call it
  good enough.


This effectively reduces the security of the hash function to that of one that only outputs 10-12 hex digits, a.k.a. 40-48 bit, for which we can easily find collisions in about $2^{24}$ evaluations which is easily doable.


  what are the chances of a "near collision"? E.g. only one or two hex
  digits are different.


As discussed in this answer the probability for at most $t$ bits difference between two hashes with $n$ bit output-length is $$p_t=\sum^t_{k=0}{n\choose k}2^{-n}$$ which means you need about $\sqrt{1/p_t}$ evaluations to come up with a desired near-collision. Strategies to finding these values can be found in this answer (including low-memory approaches).