does the $205,000 compensation limit apply to employer contributions to a 403b plan? In other words, when the employee's annual compensation hits $205,000, does the employer have to stop contributing, even if the other limits on total contribution have not been met?

Yes it does. See IRC 403(b)(12)(A)(i), which cross-references to IRC 401(a)(17). The compensation limit does apply to employer nonelective contributions to 403(b) plans.

The $205,000 limit of compensation does not apply to 403(b) contracts purchased by a church. See Code Sections 401(a)(17), 403(b)(12)(A)(i), 403(b)(1)(D), and 403(b)(12)(B). Note that church is defined more narrowly than in Code Section 415(e) although it does include qualified church-controlled organizations.