How do you know when to harvest beets?
This article will help with that: https://www.gardeningknowhow.com/edible/vegetables/beets/harvesting-beets.htm
Are these two compatible?
Yes, they are. For more info on companion plants, this article will help: https://www.gardeningknowhow.com/edible/vegetables/vgen/companion-vegetable-garden.htm
When is the best time for growing beets?
Beets normally prefer to be grown a bit on the cool side; therefore, it is better to grow them during the coolest part of spring and fall. This article should help: https://www.gardeningknowhow.com/edible/vegetables/beets/growing-beets.htm
As long as the "normal" weather is kind in your area of the world you can grow beets pretty much most of the year.
I have grown beets (in successive weeks) from March to October here in the UK and have had very good results. But (there is always a but) it depends on the variety you want to grow, your soil type and as I said the "normal" weather in your area.
Hope this helps
Walrus
Can I pick beet tops before the beet (root) is ready to be picked?
Yes, you can. This article will help you: https://www.gardeningknowhow.com/edible/vegetables/beets/what-are-beet-greens.htm
Why does my beet plant grow very nice on top but the root will not produce a bulb.
You may not have enough phosphorus in the soil, which will help with root growth. Add some bone meal to the soil. Here is more information: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/phosphorus-plant-growth.htm
I want to plant beets and turnips (fall plantings) under normal fall conditions, in northwest Indiana. Will the vegetables produce? Suggestions please.
Yes, they should produce. They are common for fall planting. You may find this article helpful:
http://www.hort.purdue.edu/ext/fallgarden.html
Can beets be grown from a beet that has sprouted?
Beet tops are alot like carrot tops. You can plant them to get seeds, which you can then replant to get full grown beets, but they will not properly grow into beet roots again.
This article give instructions on how to grow carrot tops, but the process is the same for beet tops:
https://www.gardeningknowhow.com/special/children/grow-carrot-tops.htm