Your opponents can still activate abilities, including abilities of cards in their hand (such as cycling abilities), and can still play lands.

Mandate of Peace doesn’t stop any player from casting spells in response to Mandate of Peace before it resolves.

Though other spells and abilities that are exiled won’t get a chance to resolve, they don’t count as being countered.

Ending the combat phase this way means the following things happen in order: 1) All spells and abilities on the stack are exiled. This includes spells and abilities that can’t be countered. 2) State-based actions are checked. No player gets priority, and no triggered abilities are put onto the stack. 3) The current phase and step ends. The game skips straight to the postcombat main phase. As this happens, all attacking and blocking creatures are removed from combat and effects that last “until end of combat” expire.

Any “at end of combat” triggered abilities won’t get the chance to trigger that combat because the end of combat step is skipped.

If any triggered abilities trigger during this process, they’re put onto the stack after the game skips to the appropriate phase or step.

In the rare case that the postcombat main phase is skipped, the game skips to the end step of the ending phase. If that step is also skipped, the game skips straight to the turn’s cleanup step. The cleanup step can’t be skipped.

If an effect allows or instructs you to cast Mandate of Peace outside of a combat phase, you can’t do so. Its restriction takes precedence over that permission. Similarly, once Mandate of Peace has resolved, your opponents can’t cast spells even if another effect allows or instructs one of them to cast a spell.