Will all organic appetite suppressants make your heart race?

Maybe. The best organic appetite suppressant is fiber. Not eaten separately but as a part of foods that are naturally high in fiber. This will fill you up, and keep you feeling satisfied longer. The best approach by far in my opinion. But most things marketed as appetite suppressants are stimulants of the sympathetic nervous system and will lead to racing heart and elevated bp.
Probably. Most appetite suppressants are stimulants. Caffeine, for example, is a appetite suppressant, which helps in the short term but doesn't have long term data. Whether "organic" or not, most stimulants can cause your heat to race. In the end, organic or not, these drugs should only be taken under a doctor's supervision. Make sure your doctor knows what you are taking.
Yes, try these. Water, organic vegetable broth, green leafy vegetables, natural pickles, apples are great to fill you up so you dont feel hungry and they are very low in calories.