Will simply sending base64 encoded data work?
There is no need to use base 64 encoding – this will simply increase the number of bytes you must transfer. Mobile operators normally limit mangling of responses to content types that they understand – i.e. images, stylesheets, etc.
How are the HTTP sessions handled?
HTTP sessions are normally handled either via a URL query parameter or via a cookie value. However, from what you have said it doesn’t sound like sessions are necessary.
Arbitrary sockets can be kept alive for a long time, but HTTP verbs are usually short lived. Does this mean I will need to create a new connection for each packet of data?
HTTP requests can last for an arbitrarily long period of time, just as for raw TCP sockets. A GET request can last for hours if necessary. You need not create a new connection for each request — take a look at the Connection: Keep-Alive HTTP header.
Or is there a way to send server responses in chunks, over a single connection?
If you don’t know the length of the response you can either omit a Content-Length header or, preferably, use the Transfer-Encoding: chunked HTTP header.
In what ways can an ISP proxy mess with the data, or the headers? For example, a proxy can sometimes keep a connection alive, even if the server closes it.
ISPs don’t tend to reveal the changes they make to HTTP responses. If you are concerned about this a simple solution would be to encrypt the data and specify a Content-Encoding HTTP header. This would require you to control both the HTTP client and server.