Interleaved low-energy electron diffraction (LEED) and magneto-optical Kerr effect (MOKE) measurements were carried out for room-temperature epitaxially grown FCC-like Fe/Cu(100) in the temperature range of 120-400 K. The structure of the Fe film was found to be not only dependent on thickness, as was believed previously, but also to be influenced by the temperature. Temperature-driven structural transitions were observed in the 4 and 4.5 ML films, this effect being more pronounced at 4 ML thickness. Whereas the whole 4 ML film assumes an FCC-like structure with a strong tetragonal expansion (FCT-like) at temperatures below 313 K, the bulk of the film relaxes into the 'isotropic' FCC-like structure and only the top layers remain expanded at temperatures above 333 K. Because only the FCT-like (expanded) Fe possesses ferromagnetic properties, the film becomes paramagnetic after heating above 333 K. This finding represents a new type of magnetic order-disorder transition and explains the lower value of the Curie temperature in the 4 ML Fe film as compared to 3 ML. In the 4.5 ML Fe film the similar correlation between the temperature-driven structural transition and an occurrence of the ferromagnetic long-range order was observed. Additionally, a pronounced difference in the energy positions of the characteristic maxima in the LEED I(E) curves for the (00) beam as well as a kinematic analysis of these curves imply a difference in the value of the tetragonal expansion for the entirely expanded FCC-like Fe film and the film expanded only in the topmost layers.