While "after" "afterward" and "afterwards" all have the same meaning "after" is used as part of a phrase. So, if it were "I write and after that I read." it would be acceptable otherwise it could be "then" afterward" or "afterwards". http://www.differencebetween.net/language/difference-between-after-and-afterward/
While I understand that in the purest grammatical sense you're correct here, I think the 'after vs. afterward' distinction is a bit pedantic. Colloquially, 'I write and after I read' would be acceptable in at least a few English-speaking situations I can think of.
For me, I only ended up here because of that distinction, which feels like a less valuable one compared to what I normally get out of the Greek duo lessons
Here's a link to the Cambridge Dictionary. After further searching online, I'd say that it's not used like that by the majority of speakers. As you have probably noticed in other discussions, our team is always open to suggestions. However, we cannot always include everybody's suggestions, since usage has to be taken into account. English is a language spoken by so many people that it's not possible to include every synonym or every alternative translation based on each and every personal preference/suggestion. The more important part is that we provide as many alternative sentences/synonyms in Greek as possible so that learners get to enrich their vocabulary (apart from the grammar part, obviously) :)
To me, it seems, "I write and after I read" is correct because a second "I write" is implied, as in "I write, and after I write, I read". This is similar to how "Read!" is correct, because "You" is implied. As the subject can be implied, I believe the object of a preposition can also be implied.
διαβάσω is aorist subjunctive, used after θα to form the instantaneous future (θα διαβάσω = I will read (once) / θα διαβάζω = I will be reading; I will read (several time / habitually / continually)), after να (θέλω να διαβάσω = I want to read) and in a few other circumstances.
διαβάζω is present subjunctive or the normal present indicative used for regular statements and questions.