Do all nuns wear these symbolic wedding bands? If not, is this practice limited to certain religious orders?
Rings may be worn by some members of female Catholic religious institutions or Religious Orders. The privilege also extends to Consecrated Virgins living in the world.
In 1963 the Second Vatican Council requested a revision of the rite of the consecration of virgins that was found in the Roman Pontifical. The revised Rite was approved by Pope Paul VI and published in 1970. This consecration could be bestowed either on women in monastic orders or on women living in the world, which revived the form of life that had been found in the early Church. - Consecrated virgin.
The tradition of women wearing a ring, as a sign of their mystical marriage to Our Lord as "Brides of Christ" is deeply rooted in the Church as early as the fourth century!
The plain rings worn by certain orders of nuns and conferred upon them in the course of their solemn profession, according to the ritual provided in the Roman Pontifical appear to find some justification in ancient tradition. St. Ambrose (P.L., XVII, 701, 735) speaks as though it were a received custom for virgins consecrated to God to wear a ring in memory of their betrothal to their heavenly Spouse. This delivery of a ring to professed nuns is also mentioned by several medieval Pontificals, from the twelfth century onwards. - Catholic Encyclopedia.
Generally speaking nuns of Religious Orders not governed by an abbess, may wear a ring after their solemn profession (such as Poor Clares).
Abbesses don a ring as symbol of authority over the monastery of her community. The other member of her community do not wear rings.
Like an abbot, after being confirmed in her office by the Holy See, an abbess is solemnly admitted to her office by a formal blessing, conferred by the bishop in whose territory the monastery is located, or by an abbot or another bishop with appropriate permission. Unlike the abbot, the abbess receives only the ring, the crosier, and a copy of the rule of the order. She does not receive a mitre as part of the ceremony. - Abbess (Wikipedia).