What is Biotin's relationship with Carbohydrate, fat and protein metabolism ( look at coenzymes)?

Interesting how important this vitamin is. For instance, there is evidence of Biotin supplementation improving insulin sensitivity in diabetic mice (1). This makes sense using your list of coenzymes since diabetics are unable to use glucuse, pyruvates gets converted to lactate (causing lactic acidosis if serum glucose rises too high) (similar to deficiency of pyruvate decarboxylase, 2). So instead of making lactate, supplemental Biotin will shunt the Pyruvate to oxaloacetate which takes part not only in gluconeogenesis, but also the citric acid cycle. 

Here is another demonstration of the effect of Biotin supplementation but on human subjects with non-insulin dependent diabetes (3). Noted in this study was that diabetes compared with controls had lower serum Biotin concentrations, and higher serum lactate, pyruvate and fasting blood sugar levels. Oral administration of 9mg a day of Biotin serum glucose, pyruvate and lactate normalized: "These observations suggest that the biotin administration ameliorates abnormal glucose metabolism in diabetic patients, presumably by enhancing the activity of the biotin-dependent enzyme, pyruvate carboxylase, with a subsequent promotion of glucose utilization for the entry into the tricarboxylic acid cycle" (3).

(1) Reddi, A., DeAngelis, B., Frank, O., Lasker, N., Baker, H., 1988. Biotin supplementation improves glucose and insulin tolerances in genetically diabetic KK mice. Life Sci. 42, 1323–1330.

(2) García-Cazorla, A., Rabier, D., Touati, G., Chadefaux-Vekemans, B., Marsac, C., de Lonlay, P., Saudubray, J.-M., 2006. Pyruvate carboxylase deficiency: metabolic characteristics and new neurological aspects. Ann. Neurol. 59, 121–127. https://doi.org/10.1002/ana.20709 (Links to an external site.)Links to an external site.

(3) Maebashi, M., Makino, Y., Furukawa, Y., Ohinata, K., Kimura, S., Sato, T., 1993. Therapeutic Evaluation of the Effect of Biotin on Hyperglycemia in Patients with Non-Insulin Dependent Diabetes Mellitus. Journal of Clinical Biochemistry and Nutrition 14, 211–218. https://doi.org/10.3164/jcbn.14.211 (Links to an external site.)Links to an external site.