Let me start by saying that I am not an expert in either OpenLCA or GHG emissions and that my answer maybe wrong or something that you already know but I still hope it helps.
Last question first: Different impact assessment methods use different combinations of characterization models for different impact categories. Global warming potential (GWP) is only one of many. The most recent and updated model for calculation for this impact category is the GWP100 IPCC AR6 model.
The most commonly used characterization model by the most commonly used impact assessment methods is the GWP100 IPCC AR5.
Note that some LCIA methods also use the GWP20 or GWP500 models based on the timeline.
So GWP20 is the global warming potential over a period of 20 years. Similarly, GWP100 for 100 years, and GWP500 for 500 years.
You can check the characterization factors for all methods that you have by opening the LCIA method and then going to the tab "Impact Factors". These can by sorted by impact categories.
For further impact factors for all greenhouse gases, you can refer to the IPCC website, where they comprehensively list all greenhouse gases and the impact factors associated with each for the calculation of GWPs or GTPs.
As for the greenhouse gas footprint and scope 3 emissions, these are regulated by the Greenhouse Gas Protocol.
The guidance for scope 3 emissions inventory.