I don't understand that if you have configured an IMAP client on your machine that you cannot read your downloaded mail offline. This is the whole point of an IMAP client. Otherwise you would just use the web interface...
There must be some setting that is mis-configured in your portable version of Thunderbird. I suggest you look in the settings to enable something like "offline files" or "archiving".
Someone else please jump in here if you know it.
The downloaded archive of gmail is 1 file, and a mess to try to use. I want to be able to take the mbox file from my gmail account, and view it without all the mess. Normally, when one views their email ONLINE through their webmail, everything is organized and easy to find, with a list and other things making the entire set of email, NEAT. The archive of gmail is the way to have the gmail messages on my local personal computer. TsVk!, other members can give advice if they want or know, but you still can too, if you can more clearly understand this the way I mean to say.
Really I am not too clear on ways of having other email account emails onto the local computer using programs like Thunderbird; the arrangement of where and when they go onto the computer is uncertain, something I still do not understand.