Do large- and mid-caps try to shun low share prices?
I have no expertise in this area whatsoever, but I've taken a quick look and see a significant body of research on reverse stock splits. Although there are many reasons that reverse stock splits occur, overall findings on the hypothesis of an optimal share price are inconclusive. 

Here is a review of 26 articles published about reverse stock splits in international journals from 1970 to 2010. I'll quote from the abstract:


  Reasons attributed by firms for initiating a reverse split are ‘image
  improvement’, ‘artificial increase of price’, ‘reduction of 
  stockholder servicing expenses’, ‘improving marketability’ ‘increase
  the stock's investor base’ ‘trading range hypothesis’ ‘signaling
  hypothesis’’ liquidity hypothesis’, ‘regulatory norms’ ‘facilitates
  going private’ and ‘compulsions from creditors, in case of a
  bankruptcy’. In the case of hypothesis testing, it is noticed that,
  researchers are not able to reach in a consensus on the outcome. 


Section 3.5 of this review focuses on "a widely accepted belief that forward stock splits and reverse splits are intended to keep the stock price within some 'optimal range'." Radcliffe and Gillespie (1979) "commented that reverse split cannot achieve an optimum range." Studies that focus on the Hong Kong market in particular do no support an optimum trading range hypothesis, at least not in the case of reverse stock splits. However Jain et al. (2004) "documented a significant increase in institutional ownership for stocks with prices of $5 after the reverse split." Jog and Zhu (2004) "found that institutional share holding increases after reverse split which helped them to conclude that an optimum price can reached (at least to institutional investors) through reverse split."

So while there are numerous reasons that companies may benefit from reverse splits, there is not strong evidence for the existence of an ideal share price range. If there were such a price range though, the study by Jain et al. suggests that the minimum might be around $5.