Hello,
after 24h, I’m sorry to not see an answer from Axelor team.
I think Axelor is good for your case.
Your post is to large for a precise answer from the community.
And I’m not competent for the manufacturing part (my company is just a distributor)
About API, I begin to be more expert
If you think you need an app/interface between your factory and Axelor, I suggest n8n for handle data.
(n8n is open-source, free and easy to you in docker)
(I use n8n, my post here explain to you how to have data from axelor or send and update to Axelor)
REST API with documentation : REST Services :: Axelor Documentation
Do you handle a simple curl commande for getting information from an app via API ?
The follow information depends on your version of Axelor, please remember this
I give you some information, but before you will need few hours of learn about API, cURL, and data information of Axelor 
All tables of « production »
com.axelor.apps.production.db.BillOfMaterial
com.axelor.apps.production.db.ConfiguratorBOM
com.axelor.apps.production.db.ConfiguratorProdProcess
com.axelor.apps.production.db.ConfiguratorProdProcessLine
com.axelor.apps.production.db.CostSheet
com.axelor.apps.production.db.CostSheetGroup
com.axelor.apps.production.db.CostSheetLine
com.axelor.apps.production.db.Machine
com.axelor.apps.production.db.MachineTool
com.axelor.apps.production.db.MachineType
com.axelor.apps.production.db.ManufOrder
com.axelor.apps.production.db.MpsCharge
com.axelor.apps.production.db.MpsWeeklySchedule
com.axelor.apps.production.db.ObjectDescription
com.axelor.apps.production.db.OperationOrder
com.axelor.apps.production.db.OperationOrderDuration
com.axelor.apps.production.db.ProdHumanResource
com.axelor.apps.production.db.ProdProcess
com.axelor.apps.production.db.ProdProcessLine
com.axelor.apps.production.db.ProdProduct
com.axelor.apps.production.db.ProdResidualProduct
com.axelor.apps.production.db.ProductApproval
com.axelor.apps.production.db.ProductionBatch
com.axelor.apps.production.db.ProductionConfig
com.axelor.apps.production.db.ProductionOrder
com.axelor.apps.production.db.RawMaterialRequirement
com.axelor.apps.production.db.Sop
com.axelor.apps.production.db.SopLine
com.axelor.apps.production.db.TempBomTree
com.axelor.apps.production.db.UnitCostCalcLine
com.axelor.apps.production.db.UnitCostCalculation
com.axelor.apps.production.db.WorkCenter
com.axelor.apps.production.db.WorkCenterGroup
com.axelor.apps.production.db.WorkshopSequenceConfigLine
Inside this one : com.axelor.apps.production.db.Machine, name columns are :
operatingDuration
importOrigin
setupDuration
code
stockLocation
buyingCurrency
description
numberOfStations
createdOn
machineToolLineList
archived
buyingPrice
id
brand
machineType
manufReference
processInstanceId
serialNumber
startingDuration
updatedBy
maintenanceCost
buyingDate
updatedOn
weeklyPlanning
version
endingDuration
picture
attrs
importId
createdBy
disposalDate
name
manufDate
So with n8n or cURL command
GET https://yourAxelorURL/ws/rest/com.axelor.apps.production.db.Machine/:id
yourAxelorUrl = you know this, maybe a localhost, or 192.168.1.200 …
ws/rest = this is standard for a GET - read the documentation REST Services :: Axelor Documentation
com.axelor.apps.production.db.Machine = the domain, you can change for the list a give you before
:id = you can find the ID of the machine when you are in the interface of Axelor, in edition of the machine. Just add the number of the id like 30
=> GET https://localhost:8080/ws/rest/com.axelor.apps.production.db.Machine/30
And you will have a json information about the machine number 30.
Have fun with your students !