In instances where I use the Horizontal Line drawing tool to mark the High and the Low of a Candlestick from a previous day, should I click on "Save position by screen position" to save those exact HIgh and Low levels for the future?

No, horizontal lines are automatically saved the value they are drawn on.  "Save by screen position" saves the line relative to where it appears on the screen, not relative to the price scale.

I'm guessing you would probably just want to save those horizonal lines normally without that options checked so the actual values of the high and low are used for the horizontal lines.

With that option checked, the line is going to position itself at a posiltion relative to the range on the screen. If you change the zoom or new data comes in, the actual value of the horizontal line will change.

Another question: I've noticed that in the last couple of days, the check boxes remain white (not checked) even though I check them.  So, it's confusing which box is checked. 

Formerly, I could tell which box wax checked.

Is this an issue with my TC software or a Worden issue at this time?

I haven't personnally seen it or heard of it happening. Please contact technical support. They would be better positioned to troubleshoot the issue and get it either resolved or submitted as bug to development.