In the Indian Church (the Malankara Orthodox jurisdiction) , the selection of the Catholicos is done by the Episcopal Synod, all bishops and metropolitans are members of the Synod with equal rights. We do not, as of now have the concept of a permanent/endemousa Synod.
This selection is to be ratified by the Malankara Association necessarily (the General Synod of the Church comprising of lay and clergy representatives from each parish ).
The bye-laws of the Church require the Synod headed by the locum-tenans (in case of the demise of the Primate) to convey the selection of the new Primate to the Syriac Patriarch of Antioch, and on the request of the Indian Synod, the Patriarch is to consecrate the new Catholicos as head of the Synod in India.
Since relations between the Patriarch of Antioch and the Indian Church are currently broken, the Patriarch is not invited. However in case of restoration of communion, the above system will be re-established. In the current situation, a successor to the Primate is selected when his predecessor is still living. This has to do with the ongoing litigation in the Church. The East Syrian Church also had a similar tradition of having a Natar Kursi.
Since the Patriarch and the Catholicos are the two highest ranking ecclessiastics in the Orthodox Syrian Church, in case a new Patriarch is to be consecrated, the Catholicos is to be invited by the metropolitan who acts as Locum Tenans of the Syrian Synod.
The use of lots was done once in recorded history; Cheppad Mar Dionysius IV was elected by lots in the 1830's. At that time Anglican interference had increased to dangerous proportions and out of the 4 names in the lot, 2 were said to have been pro Anglican candidates.
The selection by lots of a Primate who was thoroughly Orthodox and brought all relations with the Anglicans to an end was absolutely a Godsend.
Bishops can be selected as Catholicos and for that matter I dont think of anything that negates the selection of any monk.
There are no explicit educational requirements (however all bishops are expected to be graduates in theology/divinity at a minimum and very many have higher degrees). All bishops are monastics; as all episcopal candidates are tonsured as monks before Episcopal ordination.
I don't remember widowers ever having been consecrated as Bishops in the Indian Church, although there is nothing against it.
However within the Syriac Church, widowers are not selected to become Metropolitans and Patriarchs. I believe this unwritten rule will be followed in the Indian Church also if any such situation arises.