I don't have experience with the Google Places API, but I do have some experience with the Google Maps API and Google Fusion Tables.
When you upload a table to a Fusion Table, it can include tables with a spatial (geometry) element like KMLs or shapefiles (although, you do have to use a conversion to get a shapefile directly into a Fusion Table). Once there, you can assign symbologies and pop-up features. Fusion Tables support points, lines, and polygons.
Fusion Table layers are brought into the Google Maps API by javascript commands. You can also setup query functions for Fusion Tables using javascript. You can query pretty much any of the fields found in a Fusion Table. For example, you can have an html table of contents in which you select a query on a specific Fusion Table layer and then display the results of that query.
If you plan on creating/drawing features dynamically on a Google Maps API, I'm not sure how you will need to write your javascript to have it extract or interact with Fusion Table layers.
I'm also not sure if what you want to accomplish in some way violates the terms of use. It sounds like you want to use Google's geocoding service, but I don't know about the limitations there.
Anyway, I hope this helps to answer some of your questions. Hopefully, some others who are more familiar with the other questions can answer more fully.
Edit: Just found this link/discussion of querying Fusion Tables that may useful.
http://googlegeodevelopers.blogspot.com/2010/11/search-your-geo-data-using-spatial.html
Best Answer
Adding
output=embedto the URL will hide the left panel and search bar:http://maps.google.com/maps?output=embed&ll=54.868705,-1.593018&z=9
However, this is intended to be used for embedding purposes. As such, there is no way to reopen the left panel or search bar in the resulting map.