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But doesn't Ichiro have some influence on the hitters after him? Aside from the obvious benefit of stealing a base, doesn't the threat of stealing a base have a positive impact on the current hitter? And, how does this show up in a metric?
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Mostly, no it doesn't. It shows up in the subsequent hitter's walk rate if anywhere. The only significant influence Ichiro has over hitters after him is that they are, on average, in better run producing states (i.e. more runners on, less outs). The effect of being a better baserunner is mostly insignificant.
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Also, if Ichiro runs on the pitch and the SS or 2B has to cover does this increase the chances of the hitter getting a hit on a ball in play because of the absence of the fielder from the optimal fielding position? If so, what metric show this?
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The metric to show that would be ground ball percentage. On average, batters with higher ground ball percentage would do better in that situation. However, GIDP would likely wipe away all that and more, because it's so much more likely and so much more devastating.
I don't have time to cover all the relevant methods and stats, but if you're really interested (and you seem to be), pick up
THE BOOK. It addresses some of the specific situations you raised and is, overall, fantastic.