@Vidder Either you say Διαβάζω εφημερίδα or Διαβάζω μία εφημερίδα is correct. In Greek, the definite article and ένας/μία/ένα are often omitted for brevity reasons!!! Same thing with Εγώ διαβάζω and Διαβάζω. The -ω ending implies that the verb is 1st person singular, therefore εγώ is unnecessary and can be omitted too :)

In Greek you can use indefinite article (ένας/μια/ένα) when you're reading any newspaper just because you had nothing to do. You use definite article (ο/η/το) when both yourself and your interlocutor know the object (e.g. the sports newspaper). When both of these have no interest or importance but the important is the action, you don't have to use an article (e.g. Πίνω καφέ / Διαβάζω εφημερίδα / Φτιάχνω αυγά / Μαγειρεύω κρέας / Παίζω μπάλα). All of the above can easily respond to the question Τί κάνεις τώρα; and in fact do not need an article. This phenomenon is usual in Greek.

Yes, it is.