This plant is about a year old and has grown a lot. The branches are very flexible. They are supported by wooden dowels but without them I think the wind would cause them to break. How long until they harden like the trunk?

Varies from plant to plant. Y'all growing varieties typically take longer to harden in my opinion.

Taller, not y'all

They should produce stiff or firmer growths. But having grown them for ten years, you may be feeding them too much.
While they are not slow growers, too much fertilizer can speed up the growth process causing longer, more slender growth.

Started fertilizing in April with Hawaiian bud and bloom every other week. The plant grew all winter in a south window with a grow light. Growth rate has not changed. Growth diameter is the same as the cutting branches, which are rather thin.

I personally think that you are feeding it way too much. Mine in Florida grew thick branches and I only fed about once every six weeks. Plus there they shed their leaves and went dormant for about four months.
The leaves started to yellow and drop from October 1st on. So without active growth there is really no need to feed them. They would come out of dormancy starting about February 1st depending upon temperature. If it was a cooler winter, maybe they started mid February, late January during a warmer winter.
But once new leaves were an inch or so long, I fed them. I used an even blend of 20-20-20 at a half teaspoon per gallon.

I have a few of those that grow thin branches with big leaves. This usually occurs when I repot my plumerias with fresh dirt. Usually the branches will harden within a growing season. Staking it like you did is good for support. You can trim some of those leaves if you don't want it to get top heavy.

Check your fertilizer to see if it has high nitrogen number. For blooming choose the high phosphorus number.