The plant looks like maybe E. anoplia or a hybrid... a bunch of similar looking Euphorbias like that. The main thing at this point (as tarev emphasized along with some other good advice) is lots of light.
For years I have grown a group of plants like that (E. polygona and anoplia) on my SW-facing balcony, in day-long sun, in clay pots that dry out completely within days. In my experience they cannot get too much sun. Now we do have a very mild climate here so things may be different if you have to factor in heat. But these are rock-solid, bulletproof plants in strong light, resistant to bugs and growing with the right kind of shape (compact). There is always going to be some degree of browning of the stem (corking or whatever you want to call it) over time with long-lived Euphorbias. Nobody likes it but it's part of the aging process.
In any case strong light will make a big difference in the general health of your plant. Indoors that means hours of daily sun. Strong light will also promote the drying out of the the soil. Your watering schedule should be based around watering well and then waiting until the soil is dry (or almost dry) at depth before watering again.
If you are going to be moving the plant(s) around to take advantage of exposure, which I do regularly, just remember that big, abrupt changes in exposure are always going to be more difficult for plants to tolerate than smaller, more gradual adjustments. In other words if you want to ramp up the light, do it stepwise over a few weeks so the plant has time to adjust.
All the baby plants you managed to pull out of there look like they are in great shape to carry on.