Do Anglicans have the minor orders of subdeacon, acolyte, exorcist, lector, and door-keeper (porter), like in the Catholic Church (cf. Council of Trent ses. 23 ch. 2), and the major order of deacon? (I know they claim to have the priesthood.)
Yes, Anglicans do have deacons, and while there are people who perform the tasks of some of the minor orders, particularly acolyte, and lector, and while they often have some training in the role they serve, they are not officially sanctioned "minor orders" in the sense of being provided for in canon law. The only official orders in the Anglican church besides Deacon are Bishop and Priest.
The Anglican equivalent of these minor orders are various and do not map well to their catholic equivalents. Members of the laity can become lay readers: these are quite common in England. I can't speak of the rest of the Anglican communion.
Lay readers are capable of carrying out most functions of a priest barring carrying out sacraments/sacramental rites. Bear in mind that a number of Anglican church services do not involve any sacraments (Evensong, Matins etc.).
There are also a few of other positions which are more modern which are worth mentioning: firstly there are Ordained Local Ministers (OLMs). An OLM is an ordained person (complete with all the abilities and all the study that implies) but their work is confined to a single parish and they are generally unpaid often supporting their church work through a secular job or in my experience using a pension.
There are also Lay Eucharistic Ministers (LEMs) who are charged with the task of assisting the clergy with administering the Eucharist and with its administration to the sick and others unable to attend church.
The final position worth mentioning are the Authorised Lay Ministers (ALMs). ALMs are trained and commisioned to work in their church by the bishop (like OLMs, ALMs are only authorised to work within their parish). Generally they are specialists in a particular area (worship, pastoral care etc.) and co-ordinate their efforts with the ministry team within their parish in the area they are trained in. This is a bit less time consuming than an OLM position and has less training.
The Anglican church does have deacons. Today, most deacons are actually curates (a kind of priest in training).