As far as I know, stars (virtually all stars) begin fusing hydrogen into helium, and the rest happens based on the mass and luminosity and etc.
Out of curiosity, I was wondering: is it even possible for a star to not begin with this hydrogen fusion at all - as in, fuse possibly another element - and still survive with an average-star lifespan?
Or, is is possible for a star to form out of an interstellar cloud containing insubstantial hydrogen?
No. The (unofficial) definition of "star" is "a body supported by the pressure generate by fusion in its core". For fusion to occur a certain temperature must be reached, and most other elements have a much higher temperature and pressure requirements for fusion than hydrogen.
It is impractical to form a star from anything else, because hydrogen is so widespread. Interstellar gas is 75% hydrogen, everywhere, and it is very well mix. You simply don't get clouds of helium or neon. Even when the gas is enriched with other elements (eg from a kilonova explosion), hydrogen will still be a major component.
Stars initially fuse lithium, as the Li+H reaction occurs at a slightly lower temperature. There is a class of Brown Dwarf (with a mass of greater than 65 Jupiters) that slowly fuse their Lithium, however not at a rate that will provide substantial heat and pressure and such objects are not, therefore, stars in the strict sense. They also don't look much like "stars" glowing at most dim red.
Deuterium (an isotope of Hydrogen) will star to fuse even earlier and this releases more energy than Lithium burning. There is a class of brown dwarfs that are established for a brief period by deuterium fusion. A chemist would probably count deuterium as the same as hydrogen, but a nuclear physicist might consider them to be different species.
So the answer in the spirit that the question was asked is that every "star" starts by fusing hydrogen of one sort or another.
A large amount of light elements would be required for them to fuse, amounts not normally found concentrated together in the universe. The closest thing would be Wolf–Rayet stars that fuse hydrogen on their outer layers and heavier elements within. Also, if non-hydrogen stars existed, they would have a shorter lifespan as at the end of a stars life, it fuses everything into iron and dies.
