On your first question:
We have configured our AppExchange listings to receive a new lead on install regardless of the type of org. This is just a matter of adjusting settings on the Leads tab of the publishing console in AppExchange and associating a Campaign record with the "Send me leads on install" line. Combined with a lead assignment rule we get an email for each lead generated via trial or test drive (we've embedded demo video and therefore do not receive leads on demo anymore).
You are correct that the LMA does not create a license record for Sandbox installs, although you can still look up the install by org name in the Subscribers tab (I assume that's part of the manual process you mentioned). I've asked both AppExchange PM and other Salesforce personnel about this, since without a manual lookup in Subscribers tab it's not possible to distinguish between a successful install to a sandbox and a failed install to production (and therefore a customer who needs attention).
The answer I received from AppExchange PM was "this is a known issue. no info on whether it will be addressed." Not very helpful, but at least they know it's an issue.
Other Salesforce sources gave me some additional detail that I think is worth sharing. The LMA is trying to be smart about issuing licenses. Part of being smart is deciding which installs represent potentially paid customers. In the case of a sandbox install, the LMA interprets that install as having no possibility of being a paid license, and therefore does not create a license record. This also means that sandbox trials never expire.
A Dev org, however, is seen as a potentially paid install and therefore a license record is created and the trial expiration will be automatically enforced.
Once I knew this, we adjusted our follow-up actions with trial customers to better address the type of org their trial went to.