My understanding is that what you are asking is not possible from Google API's. Reverse geocoding inside a polygon is not a service they offer. There are some posts on other sites (e.g. https://gis.stackexchange.com/questions/22816/how-to-reverse-geocode-without-google) with the reference gisgraphy.com looking like a pretty neat reverse geocoding tool.
This still does not address your all streets in a polygon problem however. I think your only option would be to get your hands on the data (Open Street Maps) and write the code yourself. Further - if you are going to do this for a large area I would take an approach like I recommended here with grids: https://stackoverflow.com/a/18420564/1803682
I would create my grid elements, and for each street calculate all the grids to which it belongs and store in the database. Then when you search a polygon, you would calculate all the grids the polygon overlaps, and can then test the subset of road data in each of those squares to determine overlap.
I looked into this and abandoned a similar requirement a few months back and still have a desire to implement it. Most of the point/line in polygon work is happening on data created in my application (i.e. not street data) and right now that is the only data I will be including. What I am trying to say is - I hope someone gives you a better answer.
Update:
For what you are asking I still believe you will need to use a mix of your own database based on OpenStreetMap and some kind of grid analysis carried out in advance. If you have some time to commit to the project this should not be too awful to process. The database will be large, and the calculations needed will likely require a significant amount of one-time / upfront processing time. As far as highlighting routes/roads/whatever within the viewport, there are lots of way to accomplish this using the API - example here which I found useful: polyline snap to road using google maps api v3
Also useful: http://econym.org.uk/gmap/snap.htm
Note that one way streets may give some grief if using the directions api to snap to a street and you will likely have to watch for this and correct or reverse the start/end points.