Who is Shudra (or Sudra)
A Sudra is not a Sudra by birth alone--nor a Brahmana is Brahmana by birth alone. He, it is said by the wise, in whom are seen those virtues is a Brahmana. And people term him a Sudra in whom those qualities do not exist, even though he be a Brahmana by birth. (Source: Vana Parva: Tirtha-yatra Parva of Mahabharata)
A man, may be born in the Sudra caste, but if he is possessed of good
qualities, he may attain the state of Vaisya and similarly that of a
Kshatriya, and if he is steadfast in rectitude, he may even become a
Brahmana. (Source : Vana Parva: Markandeya-Samasya Parva of
Mahabharata)
This is also discussed here:
Can a Sudra become a Brahmin in 'this' birth?
Also,
[Nārada Muni:] If one shows the symptoms of being a brāhmaṇa, kṣatriya, vaiśya or śūdra, as described above, even if he has appeared in a different class, he should be accepted according to those symptoms of classification. (Source: Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 7.11.35)
From the kṣatriya known as Agniveśya came the celebrated brāhmaṇa
dynasty known as Āgniveśyāyana. (Source: Śrīmad-Bhāgavatam (Bhāgavata
Purāṇa) 9.2)
Manu also had a son known as Dhṛṣṭa, from whom another sect of
kṣatriyas was generated, but although they were born of one who had
the qualities of a kṣatriya, they became brāhmaṇas." (Source :
Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 9.2)
In addition to these nineteen sons mentioned above, there were eighty-one younger ones, all born of Ṛṣabhadeva and Jayantī. According to the order of their father, they became well-cultured, well-behaved, very pure in their activities and expert in Vedic knowledge and the performance of Vedic rituals. Thus they all became perfectly qualified brāhmaṇas. (Source: Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) 5.4.13
That wretched Brahmana who falls away from his duties and whose
behaviour becomes wicked, becomes, O king, a Sudra. (Source: Santi Parva:
Rajadharmanusasana Parva of Mahabharata)
Vishvamitra was a kshatriya king earlier. Later, he undertook intense
penance and attained brahminhood. This is discussed here.
Now, coming to your questions:
Are shudras allowed to wander near sacrifices? Can they participate in a sacrifice? Are they allowed to put a sight on a sacrifice?
In few cases No:
The Sudra is not fit for the sacrifice (the One Day Sacrifices). -Krishna Yajur Veda - Kanda 7 - p1
In few cases Yes:
ब्राह्मणान्क्षत्रियान्वैश्याञ्छूद्रांश्चैव सहस्रश:।
समानयस्व सत्कृत्य सर्वदेशेषु मानवान्।।1.13.19।।
"Let all the people from all the kingdoms, say Brahmans, Kshatriyas,
Vyasyas, Shudras be invited in scores duly honouring them." So said
Vashishta to Sumantra. (Dasaratha's Aswamedha yaga)
Vidura (born from a Sudra woman) became the disburser in Yudhisthira's Rajasuya yaga. -Sabha Parva: Rajasuyika Parva of Mahabharata
Are they allowed to touch sacrificial butter or other things?
From Santi Parva: Rajadharmanusasana Parva of Mahabharata
In few cases No:
The Brahmana is the performer or the sacrifices of the three other orders.
A Sudra, is not competent to titter swaha and swadha or any other
Vedic mantra.
In few cases Yes:
The Sudra, without observing the vows laid down in
the Vedas, should worship the gods in minor sacrifices called
Paka-yajnas.
Vidura (born from a Sudra woman) became the disburser in Yudhisthira's Rajasuya yaga. -Sabha Parva: Rajasuyika Parva of Mahabharata
However, same chapter mention that it is not true that the gods and other (Superior) persons do not manifest a desire to share the offerings in such sacrifices of even the Sudra. For, this reason, the sacrifice that consists in devotion is laid down for all the classes.
Yuga effect
In Uttara Kanda of Ramayana, the eight leading Brahmanas named Markandeya, Maudgalya, Vamadeva, Kashyapa, Katyayana, Javali, Gautama, and Narada say:
In three Yugas (Kreta, Treta and Dwapara) the Sudras were not entitled to the vitues of devout penances. The degraded caste the Sudras, having thy regeme, have engaged in austere penances. And in the Kali Yuga asceticism shall be established in the Sudras.
Footnotes
Ramcharitmanas 1.1.163 says
जनि आचरुज करहु मन माहीं। सुत तप तें दुर्लभ कछु नाहीं।।
तपबल तें जग सृजइ बिधाता। तपबल बिष्नु भए परित्राता।।
तपबल संभु करहिं संघारा। तप तें अगम न कछु संसारा।।
So, one can attain any thing by Karma/Action/Tapa:
Bali Maharaj (son of a Daitya) will become next Indra.
Vishvamitra (a kashtriya king) attained the post of Bhrahmrshi.
Bhakti of Sabri (born in low cast) is believed to be perfect.
Vidura (born from a Sudra woman) is believed one of the most intelligent persons in Mahabharata.