This is probably too late for OP, but I found a possible way to do this.
You can open the Firefox Profile folder without(no root required since Firefox owns the folder) by entering the following URL in Firefox Mobile's address bar:
file:///data/data/org.mozilla.firefox/files/mozilla/
This opens the folder thanthat stores your Firefox profiles. The actual profile folder is the folder with the random looking folder name there. In my phone for example, the profile folder is y7hj32bc.default (this name will change for every installation).
Inside the profile folder, there should be a file named:
downloads.json
You can tap on the file, and Firefox would open the file as text file. Copy the content of this file to somewhere and open it with a text editor.
The downloads.json file contains a JSON file that looks like the following:
{
"list": [
{
"source": "http:\/\/example.com\/path\/file.txt",
"target": {
"path": "\/storage\/emulated\/0\/Download\/file.txt",
"partFilePath": "\/storage\/emulated\/0\/Download\/file.txt.part"
},
"saver": {
"type": "copy",
"entityID": "%2239907-3fe8c75e5c140%22\/235783\/Wed, 17 Aug 2005 18:42:37 GMT"
},
"startTime": "2015-04-04T17:25:59.468Z",
"succeeded": true,
"totalBytes": 235783,
"tryToKeepPartialData": true,
"contentType": "text\/plain"
},
...
Your file download start date is in the "startTime"startTime field, and the file path is in "target/path"target/path.
The date is formatted as ISO8601 format.