The "strange growths" are seed pods.

The dead branches are where the flowers bloomed. It is normal for those to die if the flowers were not fertilized. The entire limb won't die. Just the twiggy looking shoot that held the flowers.

Though I can't see it all, what is showing of the Oleander in the photo looks quite healthy and happy.

not to worry! that looks like a pretty happy oleander. soon you'll probably be at that thing with pruning shears to keep it from eating the rest of your yard unless it's in a spot where it can get big and not hurt anything. I was given an oleander by my neighbor who had it in shade, so it wouldn't grow. It was a straggly 3' tall twiggy thing with two sets of leaves. Within a few months of being moved into full sun, it took off. Unfortunately, I planted it right next to my driveway, so I'm pruning the darn thing every month right now just so I don't graze it with the car every day. I'll be moving it this fall.

Now, a question for you: I take it you like plumeria? I bought one in Hawaii once, but never could get it to grow. It just shriveled up. What part of TX are you in, and what's the best way to get one started? Pot size? Soil type? Fertilizer? Water usage? THANKS!