What is the proper annotation since @SpringApplicationConfiguration, @WebIntegration, is deprecated in Spring Boot Framework?

Take a look into JavaDocs of deprecated classes:

* @deprecated as of 1.4 in favor of
 * {@link org.springframework.boot.test.context.SpringBootTest} with
 * {@code webEnvironment=RANDOM_PORT} or {@code webEnvironment=DEFINED_PORT}.
 */
...
@Deprecated
public @interface WebIntegrationTest {

---

* @deprecated as of 1.4 in favor of {@link SpringBootTest} or direct use of
* {@link SpringBootContextLoader}.
*/
...
@Deprecated
public @interface SpringApplicationConfiguration {

Is there also a replacement for TestRestTemplate()?

Yes, here it is:

 * @deprecated as of 1.4 in favor of
 * {@link org.springframework.boot.test.web.client.TestRestTemplate}
 */
@Deprecated
public class TestRestTemplate extends RestTemplate {