The early Buddhist texts describe yogic and meditative practices, some of which the Buddha borrowed from the śramaṇa tradition. The Pali canon contains three passages in which the Buddha describes pressing the tongue against the palate for the purposes of controlling hunger or the mind, depending on the passage. However, there is no mention of the tongue being inserted into the nasopharynx as in true khecarī mudrā. The Buddha used a posture where pressure is put on the perineum with the heel, similar to even modern postures used to stimulate Kundalini. Some of the major suttas that discuss yogic practice include the Satipatthana sutta (Four foundations of mindfulness sutta) and the Anapanasati sutta (Mindfulness of breathing sutta).
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.
In 2009, the Council of Ulemas, an Islamic body in Indonesia, passed a fatwa banning yoga on the grounds that it contains Hindu elements. These fatwas have, in turn, been criticized by Darul Uloom Deoband, a Deobandi Islamic seminary in India. Similar fatwas banning yoga, for its link to Hinduism, were issued by the Grand Mufti Ali Gomaa in Egypt in 2004, and by Islamic clerics in Singapore earlier.
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.