Care for Modal as you would cotton.
It can be washed in any water temp., follow label directions.

Full strength chlorine bleach can weaken the fiber. Dilute solutions can be used safely for stain removal and whitening. However, even dilute solutions will weaken fibers causing them to rip and wear out if used too often.

Yes, I think LCB would harm this fabric blend. I would never use LCB (liquid chlorine bleach) on any fabric that has rayon and/or spandex(lycra), except as a Last Resort and I was ready to toss it anyway in case of fatal damage.

If you must, try a non-chlorine bleach instead, like Clorox II or OxiClean.

I would also recommend washing the colors on a COLD setting.

I have some rayon/spandex tees from Talbots, and that stuff will pill in no time flat. Best to wash all of them on a Delicate cycle. Be aware of that with the rayon blend. Knits with rayon and spandex generally don't hold up well. Maybe the cotton in your DD's will help.

I have those t-shirts as well and I've found them tricky to launder. I started them on warm but didn't like the result. I use cold, dry them partially, then lay them flat before folding.

If they are being washed with liquid bleach they will last one season before they begin to look old.

Many people still use clorine bleach but I don't see the necessity. Persil with a small oxyclean booster even gets white dishtowels bright and stain-free.

Try putting your shirts in a large-size lingerie bag to reduce friction that can cause pilling. Get a good one - Woolite brand bags are good. The cheap ones have a too-large weave and things get caught in them, defeating the purpose.