Hello Johan, and welcome to the LiveCode forums!
reading and writing text files
LiveCode can definitely do that. There is a lesson in the LiveCode lessons website which shows you how to do that: http://lessons.runrev.com/s/lessons/m/4 ... put-output
manipulating images (scaling, cropping, resizing, putting a logo on it)
Unfortunately I have very little experience with images and LiveCode. It does seem possible for LiveCode to perform some of the effects that you mention. Take a look at the section in LiveCode lessons on images: http://lessons.runrev.com/s/lessons/m/4071/c/16770
publishing all this to desktop Linux or Windows
LiveCode is very good at cross platform. It has the ability to allow you to create your app once, and publish it on Windows, Linux and Max OS X with little to no changes. It also has the ability to allow our to create and iPhone or android app form your code with some minor changes.
As for your other requests, I am afraid that I do not know. You should take a look in the LiveCode lessons (http://lessons.runrev.com/s/lessons
), as it covers a wide range of topics. I see it has a few lessons on working with databases. This may help you with your request for reading and writing databases.
Anyway, good luck in your search for an alternative to Director/Flash.