Can I download older versions of arcmap?
I would recommend contacting Customer Service to see if this is possible; https://www.esri.com/en-us/contact#c=us&t=3
Now you can import your old .mxd files into ArcGIS Pro and use them that way, if you would like. That is what I would do.
Here is some steps on how to do this: Import an ArcMap document—ArcGIS Pro | Documentation
In the title of this thread you mention ArcGIS Pro, and then in the description you say you want to download an earlier version of ArcMap. Are you concerned with the compatibility of an ArcMap .mxd within ArcGIS Pro? If so I believe George addressed that question adequately with the provided documentation.
As for your second question about downloading an earlier version of ArcMap, this can be done through within your MyEsri page. I will summarize this process and also provide some documentation that will include some useful visuals.
Log into your MyEsri page, click the My Organizations tab, then the Downloads button just below it. This will bring up a listen of products, you can specify the version you want, hit View Downloads, and then download the desired software. The documentation for this process is located at the URL below.
How To: Download and install Esri software from My Esri
I hope this helps!
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