Membrane is synthesized in the Endoplasmic reticulum and Golgi apparatus, along with membrane-bound proteins and packaged into vesicles. The vesicles fuse with the membrane via SNARE proteins. I don't think plasma membrane is produced without any functional proteins attached or contained (e.g. mucus secretion), and I'm not sure what controls the 'mature size' of cells. Various extracellular growth factors can control the growth/division/apoptosis behavior by promoting synthesis and inhibiting degradation of proteins (so-called mitogens and growth factors), and there have been experiments showing that adherent cell types will undergo apoptosis if they cannot spread out to some minimum size (IIRC, O'Neill, Jordan, and Ireland, Cell 44, 1986). Muscle cells can grow very large and are even multinucleated. My (cultured) cells experience contact-inhibited growth- once they touch, they stop growing.
There's a breed of cows selected due to a deletion mutation in myostatin- their muscles grow abnormally large. That's control of the organ/tissue size, not of the individual cells, but it may be a place to start.