Yes the Mennonites do grow quite a lot of corn in the country, both for local consumption and export. Corn in the summer, and beans in the fall and winter is the norm, with some sorghum in either (more in winter though during bean cycle). The larger mechanized mennonite communities plant mainly hybrid feed corn - as opposed to sweet corn for human consumption. Some of the horse and buggy old order Mennonites, plant heirloom varieties. Belizean smaller farmers, referred to commonly as milpa farmers, farming from 1 to 5 acres, more often plant open pollinated varieties, from which they can save the seeds for replanting. These are not sweet corn - they are either yellow or white types of corn, and occasionally purple corn. The white type is preferred for tortillas. Cardi and ministry of agriculture develop and propogate these seeds especially for use in our climate, by smaller belizean farmers.
Yes all of the crops you mention will grow here, many have seasons, preferring either the wet or dry season. The district agriculture officers of the ministry of agriculture (mnra - ministry of natural resources and agriculture) can provide you with more information when you arrive in country. Information available from them about organic growing as well. Growing is easier in the tropics - but that also applies to the weeds and pests, who also do better down here.
Heirloom seeds are generally not available commercially here - but you will likely find a network of seed savers and traders. There are more organic folks i would say located in southern belize - toledo district than any other one district. Cayo and orange walk have much commercial ag. There is a wonderful ngo operating in toledo dist. For close to 10 years, called sustainable harvest international. Their website is sustainableharvest.Org wonderful group - they host the annual organic fair for the country, held in punta gorda, toledo.
I have never heard of anyone growing grains in screened greenhouses - remember, many of these plants rely on insect pollination services. Lettuces and tomatoes are sometimes grown screened. Citrus nurseries must be grown in screened houses, to prevent spread of the dreaded citrus greening disease.
Good luck on your search. It is not paradise here, but many of us would not live anywhere else. Find in the signature below website for the belize ag report. It might have some articles of interest to you. It covers both commercial, home and recreational agriculture. The current issue (issue 26, pg 18) has a page of charts of agricultural trends in spanish lookout, cayo district over the past decade.