The forms चाहता, चाहती and चाहते etc always mean 'want'. It is only the form चाहिए that indicates modality(should). One additional note is that चाहिए may also sometimes be used to mean 'need' (rather than 'want'). You can differentiate between the two uses of चाहिए based on context. I will give you examples of all three cases:
मैं जाना चाहता हूँ - I 'want' to go
मुझे जाना चाहिए - I 'should' go
मुझे इन्साफ चाहिए - I 'need' justice
पढ़ने is the oblique form of पढ़ना. It is used when there is a postposition following it. Eg: नेहा पढ़ने में दिलचस्पी रखती है - 'Neha is interested in studying'.
It is also used with आना and जाना because of an implicit postposition ('ghostposition') between आना/जाना and the infinitive (पढ़ना) which functions as the object of आना/जाना. Eg: नेहा पढ़ने जा रही है - 'Neha is going to study'.