The box width is set to 100% by default, same as in the lite version. However if the container element width is set differently, the percentage width can be small. On the Theme Options -> Overall Box layout panel you can change that wit the “Search box width” option. Try pixel values like 300px or 400px.
I've checked the code regarding the custom field search and the version log as well. The only known issue with the current version is when only custom fields are selected to search for, the search doesn't trigger. If this is the case and you only want to search custom fields, then a bypass solution is to enable at least one other field to search as well. I suggest enabling the “Search in post excerpts?” option on the General Options -> Behavior panel. Post excerpts are usually empty, and it won't affect the pure custom field search that way.
If however this is not the case, I suggest updating the ticket information with log-in and FTP details, so I can investigate if there is an unknown issue there.
Image Problem: It's not necessarily that. The plugin gets the images as it finds them, there are not modifications to the URLs. In your case the problem might be (I'm guessing) is that you might be using a plugin to enforce HTTPS over HTTP when pushing from test to live environment. So the images might be stored with a HTTP url in the database, but later filtered to HTTPS urls. If this is the case, the following custom code might help:
Try copying it into the functions.php file in your active theme directory (copy from line 3)
This might not do anything if the issue something else, but it's worth a try.
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