Hi! Sure, I mean… sure u can! :v
I will try to adress each point –for most of them at least– of your detailed question regarding the Editor and Wix Pro Gallery:
The ability to download images/videos to a device
You can do so by toggling on this setting (highlighted) in the Options menu of the app settings. I am not sure if this allows to download the video files, but at least I know for sure it works to download the image files.
Videos in your gallery can be set to play automatically, onClick or onHover, with sound on or off, among other particular settings. Now, these settings only modify the display of videos in the gallery’s main layout, not the
expansion mode . These options are available in Settings > Advanced.
Images/Videos should have a Title and a Description
Indeed. In fact, depending on the default settings, the gallery app doesn’t leave an image without a default name and description that you can later change (there are two ways I can think of doing this; but it depends and… keep reading).
To actually show each image’s Title and Description you need to mark both checkboxes (‘Title’ and ‘Description’) at Settings > Layout > Text.
Depending on the settings above, titles and descriptions will be shown in all images or just the hovered one, above or below the image or even covering it, and you can of course customize their style formats (font, size, color).
The Image/Video can be tagged to group similar files
This is a tricky one. To do this and be able to perform them according to their custom tags, I would recommend to use your Content Manager and create an image database , then to connect your pro gallery to it through a dataset .
Once you create your database, you’ll have to name and administrate (and optionally arrange for better comfort) your fields per item; I assume for what you’re asking you’ll need at least these fields: Title (text), Description (rich text), Image and Tags. You’ll be able to filter your items by their tags through the connection.
This is a very different process than that of manually adding your images to the gallery the “normal” way (‘Manage items’), but gives you that and more capabilities and, although it may sound a bit
techny , doesn’t involve code (or not necessarily). Among the benefits of using an image database is the ability to upload new items (your images) directly from your site’s manager without the need to open the editor, and these changes are visible on your live site instantly.
Although I do must warn u that you’ll have to learn the differences between Sandbox & Live . I have to warn you too, that –as I sense– if you want to display all images in your gallery and let your user filter them by tag live in your site you do will need some coding through Velo, but as I remember it is possible and not extremely complicated (others have achieved so too) (I may be wrong tho).
The ability to categorise images/videos into Folders/Albums/Groups
Maybe Wix Pro Gallery isn’t the app for this; if you’d want to display your images grouped in folders or albums then –and I didn’t know it either before but– there is an app for that: Wix Photo Albums . Try that one instead if albums are what you want to achieve, but I couldn’t tell you for sure if it’s available in EditorX since many of the Wix Apps aren’t \:
The ability to search the images/videos using Site Search (Well this feature is indeed not available for EditorX but we all hope it gets added soon)
Does Wix Pro Gallery let us upload images/videos from the Wix Mobile App?
Yes , since it has a dashboard that allows you to manage your apps and their contents. But most probably if you have uploaded your images traditionally and not through databases (not sure); then again I couldn’t tell if the content manager is available from mobile.
Does Wix Pro Gallery let us upload and download videos and images in the highest quality?
It downloads them in the same size, format, quality & resolution as originally uploaded. No reductions as far as I have understood and tried out myself.
Nonetheless, it reduces their quality onDisplay (when they are shown in the live site without downloading) with the best compression possible not to affect quality nor page-load speed. This percentage, however, can be manually changed in the app settings:

Does Site Search let us search images/videos in the Wix Pro Gallery?
No. Even when available –which isn’t yet–, Site Search worked pretty poorly every time I tested it in the classic editor.
Now, if you’re talking about Search Engines (like Google/Bing/Ecosia), Wix does indeed allow your galleries’ contents to be found and indexed by them; but for them to appear in search results efficiently… well that’s another story, since Wix’s SEO is quite good for Premium sites but barely existent for Free ones ':
Is there a limitation of file size while uploading images/videos to the Wix Pro Gallery?
Read This Article and look up for ‘File Size Limits’ column on the right of the table.
I hope I’ve helped somehow to solve your doubts 
P.S. - sorry all my screenshots are in Spanish ':l