How effective to use zinc strip to prevent roof moss

Nail them down, the same way your shingles are nailed down. That is, you nail down a course of shingles, then lay down the new course above the last course, working up the roof. This way the next course hides the nails holding down the previous one. At the ridge, before a ridge cap goes on, you also nail down the zinc strips, placed so they will protrude beyond the ridge cap. Finally, the ridge cap goes on. Enough of the zinc protrudes so that the metal gradually leaches out with every rain to impede moss accumulation.

The bleach mix will be dilute enough that it will not be of harm. I've used it myself at least a few times to kill off algae that has accumulated.

You can also use things like Oxy-Boost and Stain Solver, a bleach substitute which I believe contains some hydrogen peroxide. It will also work against light moss and algae, yet perhaps be less harmful to plant life underneath.

More importantly is to fix the source of the moss. Roofs that gather moss tend to be too shaded, under big trees that stop it from drying out. You may need to open things up, just by a bit, but enough to let it dry out. Yes, those trees are a natural source of air conditioning, but trees tend to grow. So trim off branches that overhang the roof. Cut a few others out to let the sun in.

For roofs that have only a bit of green, you may find it is sufficient to add zinc strips along the top. The zinc leaches out, keeping the roof clean of moss accumulation. There are also shingles with a moss inhibiting additive that prevents moss from growing. Copper strips should also work, and copper flashing is easily available. I've even read that you can glue (use silicone) copper pennies just under the edge of the top row of shingles. They are mainly zinc anyway.

If the moss had accumulated heavily, the problem is your roof likely needs repair, as it will have been too wet for too long. The shingles will need to be stripped off and replaced, but probably the roof decking will be soft and need some repair too. This can best be judged by a professional who will inspect the roof carefully.