I don't get pulled over often, but the last time that I got pulled over I told the officer that I believed I was going the speed limit. It's a subtle distinction, but it seemed to work as he had no other reason to pull me over and didn't write me a ticket.

The last time I tried that the officer angrily barked back at me, "GIVE ME A NUMBER." So YMMV, I guess.

What if you really are doing 35 and tell the officer "I was doing 35"? Surely that's perfectly fine right?

No. You're required to not exceed the speed limit.

I suppose it depends what is the intended meaning of the question - you want my average speed over the last few minutes? Since you've been following me? The latter is probably the intended meaning and, in that case, I really can't be expected to know.

A driver obviously cannot constantly keep his/her eyes on the instruments.

No obviously not, but you should have a sense of speed enough to not deviate much from the last time you checked.

Not precise enough that I would testify against myself or against someone else.