What are the white spots.

Last summer there were aphids and white fly. There's much powder from honeydew.
As weather permits, I'm resizing the landscape plants. I'd like to shred the waste and feed the compost pile.

I'm finding lots of these white things on much of all plants even though we've had freezing nights.
1. How to rid the plants of these white "bugs"?
2. Is it smart to mulch the cut-off portions and add to a compost pile... or would doing so just be spreading the problem?
3. If I make a separate pile of the mulch, will letting the pile die and sit for a year kill off these bad bugs?

I dislike throwing stuff away.

Pictures 3 and 4 look like a bacterial leaf spot.

If you want to compost this material at all, use a HOT compost. Absolutely do not just shred or chip and mulch with the material.

Not sure about the white spots. Fungus??? Pictures are blurry on my monitor.
Do these things move?
Sedentary (or almost sedentary) insects include Woolly Apple Aphid, Meally Bug and certain Scale insects. You might look up pictures of these and see if that matches what you see in your garden.

Aphids and certain other insects that secrete honeydew usually move around on the plant, but the honeydew is all over, drips on the ground or anything under the trees. It attracts yellow jackets and a type of mold called Sooty Mold. Makes the plant look black. Sooty Mold is not inside the leaves like your pictures 3 and 4. Sooty mold can be washed off.

Composting known disease carrying plant material:
Depends on the disease.
Some diseases are so strong they stand up to cold composting methods and re-infect the plants. A carefully controlled hot compost might be used.
Some diseases are all over the place in such quantity that hot composting (even if you miss some of the disease organisms) is probably fine. Even if you threw away the diseased parts more would blow in from next door or down the street, so go ahead and compost and re-use.
Some diseases are not very common, and properly disposing of the infected material is the best way to get rid of it. It probably won't come back.

White Wax Scale is as close as I find pictures of.
These were found on my gardenia and English laurel the first 60 degree day we had right after several night lows in the 20s.

White Wax Scale, is it wrong to squish them? When I try picking them off, many pop open and slightly sticky juice runs out. Does that harm the plants?

General life history of many scale:
Hatches from eggs that are either hidden in the cracks in the bark or else protected by the mother scale (varies with the species).
Babies crawl a short distance, then settle down and grow a hard, protective shell. Not all species have a very hard shell, some are somewhat soft.
Some species have only one generation per year, some have more, especially in warmer zones.
They suck the juices out of the plant.

Control:
Eggs and crawlers can be smothered in the right season with oils, and soaps may work on some.
The adult, with the waxy, harder shell is protected from pesticides that just hit the surface. They can be poisoned with systemics.

You can physically control them by scraping them off, or popping them as you are doing, if there are not very many. This does not hurt the plant. Watch for more, the growing babies. Hand control is a long term project. Babies are quite hard to see, so keep a sharp eye out and start squishing as soon as you see them, hopefully before they produce babies of their own. If you are willing to try pesticides, you can start with milder products like insecticidal soap or oils. Follow the label directions and make sure the plants you want to treat are listed.

Here is some info about White Wax Scale:

http://www.aphis.usda.gov/plant_health/plant_pest_info/pest_detection/downloads/pra/cdestructorpra.pdf
http://caps.ceris.purdue.edu/webfm_send/1553

I would take the plant part with live pests to your local department of agriculture office for proper ID. This pest is not currently (2011 data) known in the USA, but has been intercepted in fruit and other plant material entering the USA. It has a high probability of surviving and becoming a major pest, especially in the southern tier of states.