If you have high blood pressure, diabetes, or heart problems, ask your doctor what you can do. You may need to avoid certain postures, like those in which you're upside down or that demand more balance than you have right now. A very gentle program of yoga, coupled with a light aerobic activity like walking or swimming, may be the best way to start.
Ascetic practices, concentration and bodily postures described in the Vedas may have been precursors to yoga. According to Geoffrey Samuel, "Our best evidence to date suggests that [yogic] practices developed in the same ascetic circles as the early sramana movements (Buddhists, Jainas and Ajivikas), probably in around the sixth and fifth centuries BCE."
Weight training aims to build muscle by prompting two different types of hypertrophy: sarcoplasmic and myofibrillar. Sarcoplasmic hypertrophy leads to larger muscles and so is favored by bodybuilders more than myofibrillar hypertrophy, which builds athletic strength. Sarcoplasmic hypertrophy is triggered by increasing repetitions, whereas myofibrillar hypertrophy is triggered by lifting heavier weight. In either case, there is an increase in both size and strength of the muscles (compared to what happens if that same individual does not lift weights at all), however, the emphasis is different.
Core: Yes. There are yoga poses to target just about every core muscle. Want to tighten those love handles? Then prop yourself up on one arm and do a side plank. To really burn out the middle of your abs, you can do boat pose, in which you balance on your "sit bones" (the bony prominences at the base of your pelvic bones) and hold your legs up in the air.