Final answer:
The events in question occurred during the 'Revolution and Republic' period, including the Texas Revolution, the creation of the Lone Star Republic, and the subsequent Mexican-American War.
Step-by-step explanation:
The era during which events related to European power trying to control Texas missions, the establishment of towns, and the creation of the Republic of Mexico took place is known as the Revolution and Republic period. The Texas Revolution started in 1835, ultimately leading to Texas declaring independence from Mexico in 1836.
This resulted in the formation of the Republic of Texas, or the Lone Star Republic. The Republic of Mexico was created after Mexico gained independence from Spain in 1821.
The American settlers, driven by the notion of Manifest Destiny, continued to influence events that eventually led to the annexation of Texas by the United States and prompted the Mexican-American War between 1846 and 1848, further affecting territories like New Mexico and California.