The use of a low annealing temperature during the production of coils made from superconducting materials is very important because it reduces the production costs. In this study, the morphology, transport critical-current density (Jc), irreversible magnetic field (Birr), and critical temperature (Tc) of straight wires and small 2% C-doped MgB2 coils were investigated. The coils were made using the wind-and-react (W&R) method and annealed at various temperatures from 610 °C to 650 °C for 2–12 h. Critical-current measurements were made for both the coils and straight wires at the temperatures of 4.2 K, 20 K, 25 K, and 30 K. During our research study, we determined the process window that provides the best critical parameters of the coils (annealing at a temperature of 650 °C for 6 h). Moreover, we observed that small coils made with unreacted MgB2 wire and then annealed had morphology and critical parameters similar to those of straight 2% C-doped MgB2 wires. Moreover, small-diameter bending of 20 mm and 10 mm did not lead to transverse cracks, which can cause a large reduction in Jc in the coils. This indicates that the processes of optimization of thermal treatment parameters can be carried out on straight MgB2 wires for MgB2 superconducting coils.