Is it necessary to enter into a state of Ihram to enter the Haram and pray in the Mosque for someone who is within the Meeqat (in Jeddah, for example)? And what are the temporal conditions for visiting the Sacred Mosque?

Whoever resides within the miqat boundaries may enter the Sacred Precinct (Haram) and pray in the Sacred Mosque (Masjid al-Haram) at any time, even if not in a state of ihram. If they intend to perform Hajj or Umrah, they should enter ihram from their place of residence, and it is not obligatory for them to go to the miqat. This is based on the words of the Messenger of Allah, peace and blessings be upon him: "These are for them and for those who come upon them from other than their people, intending Hajj and Umrah. And whoever is within that [i.e., closer to Mecca], then from where they commence, even the people of Mecca, from Mecca." Visiting the Sacred Mosque and undertaking a journey to it is permissible at all times, unless there is an impediment like al-Nuzul (descent/illness that prevents one from performing rituals).