Below are my comments:
1. It varies from firm to firm, but normally what I have observed is, PP team does the TECO of the order and CO team does settlement but there is no hard and fast rule for the same. As far as SAP goes, yes CO team has to do it.
2. For settlement - tcode is KO88 for individual orders and CO88 for collective settlement.Closure can be done through CO02 - Go to functions-restict processing-Close
3. For Archiving of Orders - Use CO78 - Maintain the variant along with the order numbers. Refer sdn link as well for additional inputs - Production order archive
4. Normally for any firm to settle an order, it takes 3 days of the next period. Meanwhile the production representative can keep the order open and book the components and GR by keeping the previous posting date. Moreover you can also do TECO if mandatory but continue to post cost and consume components and activities as per the needs.
As a PP representative - Normally a guidelines needs to communicated to the team that what is the period by which standard cost estimate would be run or else they can anytime analyze the material master and understand from the costing tab 2- what what the previous period, current and future period rates to understand the frequency at which cost of the materials gets updated.
Hopefully it will help.