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I found this in TRP-03-114-02:

“The concrete used for the culvert’s top
slab and curb consisted of a Nebraska 47-BD Mix with a minimum compressive
strength of 31.03 MPa (4500 PSI). The actual concrete compressive strength for
the culvert’s top slab and the curb on test day, as determined from concrete
cylinder testing, were found to be approximately 6992 MPa and 41.64 MPa (6039
PSI), respectively.”

From my understanding our structures department
designs concrete using f’c=3,500 psi. 

For the adhesive option when the slab is 10”
what is the minimum f’c?

For the bolt through option is there a
minimum slab thickness and what should the minimum f’c be for the slab?

With the through bolt option, the thickness of the slab shouldn't make a difference to the attachment strength.  I say this assuming you do not have slab thicknesses less than 8" (the minimum found during a recent review of state DOT culvert standards) and you utilize the bottom washer plate on the under side of the culvert top slab.

Report TRP-03-278-13 recommends a minimum f'c of 4,000 psi for use with the epoxy anchorage option (8" embedment depth with a minimum epoxy bond strength of 1,300 psi).  Changes to the minimum f'c could be dealt with by variations of embedment depth and/or epoxy strength.  Using the design procedure for epoxy anchorages found in ACI-318-11, you would first calculate the strength of the recommended anchorage design, then lower the f'c and adjust the embedment depth and the epoxy strength until the estimated strength is equal to or greater than the original design. 

If everything else was the same:

What depth of embedment would I need with the same bond strength and f'c=3500 concrete?

What type of bond strength would I need with the same embedment and f'c=3500 concrete?

using the ACI-318 procedures by the book: a concrete with f'c = 3500 requires 8.5" of embedment with an epoxy strength of 1300 psi.  The failure mode is calculated as a concrete breakout failure, so 3500 psi concrete with a stronger epoxy will not produce an equivalent anchor strength to the 4000 psi concrete without a deeper embedment depth.

In report TRP-03-264-12, some dynamic loading coefficients where proposed to account for the increase in material strengths during a dynamic loading event.  Note, the ACI code was written for statically loaded connections.  Using these coefficients for steel, concrete, and epoxy strength under dynamic conditions, the failure mode for the anchors is epoxy bond failure for either concrete strength, f'c = 3500 psi or 4000 psi.  Thus, if you wanted to use these dynamic coefficients, you would not need to change anything for a 3500 psi concrete slab.

For the bolt through option, what is the f'c that the concrete was designed for?  Or does that not matter because the loading is going to be dynamic as well?

The mix was specified as a 4,500 psi concrete mix.  However, utilizing a 3,500 psi mix will probably have minimal effect on the performance and damage.  Make sure you have a minimum slab thickness of 7" (the thicker the better for weaker concrete), the slab has adequate reinforcement (similar to the as tested culvert slab - shown in TRP-03-114-02), and use the large bottom washer plate to distribute the load out over a wide area.