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The answer in short? Yes.
But why till? Save time, money and backache...try lasagna gardening. Once you do, you'll never touch a tiller again!

The mushroom compost will be an exellent addition! For a nitrogen boost, I've always added some alfalfa to my 'recipe'. Look for it at any feed supply store. I used a 50lb bag on a 1000 sq ft garden.
If you have any more questions, feel free to ask~
Happy gardening!

I would not put too much wood ash directly onto the garden. Small amounts are ok, but too much will "burn" the ground. When wood ash gets wet, it leeches off lye. Lye is caustic and too much will kill the plants. I would add the wood ash to a compost pile to mature it a bit and let the lye mellow a bit.
But, if it is only a few shovels full and spread over a wide area, that will be fine.

This article will help you:
https://www.gardeningknowhow.com/composting/ingredients/wood-ashes-compost.htm

Sulfur is good for the pH. This article has a few other ways you can consider for lowering the pH:
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/raise-acid-level-soil.htm

Bone and blood meal are great, but just realize that the nutrients in them are not immediately available in the soil after applying. It takes some time for them to break down. Composted manure can get the nitrogen in the soil very quickly and will also help plants with taking up phosphorus already in the soil. I would make sure to use a bit of the composted manure with the blood and bone meal to get good balance and timing.

Adding compost is the quickest way to amend the soil. This article will help you: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/amending-sandy-soil.htm

This article will help with amending the clay soil:
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/how-to-improve-clay-soil.htm

You would probably find lasagna gardening helpful in amending the very heavy soil: http://web.extension.illinois.edu/coles/yg/060309.html