The best option is probably to replace the logs that appear to have rotten ends so they don't lose support on the underlying logs. To do this, it might be easier to dismantle part of the deck to access them and then lift the frame of the deck.
In connection with that, it might be wise to add more layers of logs to create a level platform for the deck construction.
How old is the stone crib, and how long/when has it been sinking? Do you think it has stopped sinking? That might influence how you choose to proceed.
Regarding fastening materials, it depends a bit on where the dock is located. Is it on a lake, the east coast, or the west coast?
If it's the east coast or a lake, hot-dip galvanized should be "enough" for quite a long durability without breaking the bank, but otherwise, acid-resistant steel is the way to go. I assume the west coast is more aggressive towards steel, so there, acid-resistant might be necessary, though I've never built docks on the west coast.
As for wood selection, pressure-treated wood for contact with water is likely the best choice. The old utility poles that were often used in the past, which were impregnated with creosote, are not very environmentally friendly or good for handling yourself (and also hard to get ahold of

).