The first law states that energy can neither be created nor destroyed; it can only change form. Photosynthesis is an example of this. The energy of sunlight is converted via photosynthesis to energy stored in the chemical bonds of the molecules making up the plant.
As stated by the expert: The First Law of Thermodynamics IS the Law of Conservation of Energy. The Second Law of Thermodynamics is not directly related; however, it provides additional restrictions, as to what can, and what cannot, be achieved.
Not exactly. The first law of thermodynamics, i.e. the law of conservation of energy, also accounts for heat as one of the many forms that energy can take. There is no one law called "the law of thermodynamics", but there are several "Laws of Thermodynamics" (note the plural form "LAWS").
The first law of thermodynamics is that energy can not be created or destroyed but it can be changed from one form to another. Cells USE energy, but do not create it themselves. So yes, they do follow the first law of thermodynamics.