We stayed very close to pier 39 recently. Instead of using a taxi for sight seeing we bought the three day city bus pass. The buses are accessible, my husband is in a wheelchair, and we had fun hopping on and off at all the tourist attractions. Some of the steep streets were a problem.
I don't know about chart house but I was there recently and ordered an Uber which picked me up right next to the pier in front of the aquarium. I would still check around though.
Not close. The taxi would drop you off on Embarcadero the chart house is towards the back of the pier.
I don't know exactly but in general the taxi's can't get very close to the shops and restaurants on the Pier
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