How to nail furring strips without them cracking

First concideration, I would encourage you to use a T&G roof sheathing such as AdvanTec instead of standard CDX plywood. This will give you a better fit and not have to worry about spacing gaps in the plywood. The price is comparable, so there is no cost advantage to using regular plywood.

The sheathing should be secured with 6d galv ring shank nails at apx 8" spacing. 1 5/8" coated drywall screws are great, but more time consuming to install. Don't use non galv common nails, as they tend to back out after a few years.

If your sheathing fits properly on the joists with 3/4" overlay, nailing straight down is fine as long as you bite into solid wood. A slight angle towards the joist is OK as long as the heads of the nails or screws don't protrude above the sheathing. Protruding nails can damage your roofing material, assuming you are not planning on using a layer of Grace Ice and Water shield under your final roofing materials.

I'm a little lost. First, what are you using to attach the furring strips to the concrete? I'm hoping some construction adhesive and a few concrete screws/anchors to bear the weight You could use the same thing to attach the Hardee-board directly to the concrete (with a layer of moisture barrier separating them to keep moisture from the concrete seeping into the siding). I'm also not sure why you're telling us what the garage door is framed with if these 2x6s aren't a substantial part of the anchor for the furring strips.

Anyway, assuming it cannot be done any other way, I'd say that using 1" nails where 1-1/4" is recommended is acceptable; maybe use a few more along each furring strip to compensate for the decreased "bite" of the shorter length of each nail. Understand that a pneumatic nailer will drive the nails down beneath the surface of the Hardee-board, so even with 1/16" clearance that 1" nail may contact the concrete. The only additional recommendation I would give is to nail in at a slight downward angle where possible, as if you were hanging a picture. The angle will not only give you a little extra depth to drive the nail, but the angle will increase the leverage the nail has to hold the siding, like using a nail to hang a picture.