All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.
According to the majority of the scholars, the person who must perform ablution for each prayer is the one who suffers from urinary incontinence, continuous vaginal bleeding outside the menstrual period, and the like. If you suffer from a similar condition, then you are obliged to perform ablution for each prayer according to the majority of the scholars. However, the Maaliki scholars held that the one who suffers from any form of continuous incontinence does not have to perform ablution for each prayer and that his ablution is not invalidated unless he invalidates it intentionally. This view was also adopted by Shaykhul-Islam Ibn Taymiyyah .
If your situation is as described in the question and you want to attend the 'Eed prayer in the Haram and it is difficult for you to go out and perform ablution when the time for prayer is due, then you can follow the Maaliki school in this regard, and there is no harm in performing the 'Eed prayer with the ablution performed for the Fajr prayer in this case.
The same applies to the Friday prayer. If you performed ablution before the Zawaal (i.e. the time when the sun moves away from its peak and this is when the time of Thuhr prayer starts) and went to the Haram, then that is sufficient and you are not obliged to perform ablution again according to the Maalikis and also according to the Hanbalis who held that the time of the Friday prayer starts when the sun has risen the length of a spear. You can make use of the concession in this case given the need for it, and there is no harm in that.
As for you second question, please refer to fatwa 330157.
Allaah knows best.