Yes, you definitely can. Although I see basically no practial application for that ;).
A bit more specific: "eine/einen" always defines, that the count of that object is one, but an unspecific one. You don't know if it's the new york times, sun, or whatever newspaper there might be. To say "one newspaper" would sound to me like if you wanted to express that you only took one of a pile of the same newspapers (because you could take more, if you wanted to bring them to your friends).
When the verb is in the singular form, it's "she". When the verb is in the plural, it means "they". Sie hat vs. Sie haben = she has vs. they have.
On a side note, Sie haben can also mean "you have" in singular and plural in the polite address: Sie haben da eine Nudel im Gesicht means, is spoken to someone you have a noodle there on you face (which btw. is the start of an incredibly funny video from a German comedian from the 80's). Context is everything.
German is pronounced the way it is spelled. The letters are not always pronounced the same way they do in English though. Take a look at this chart - http://german.about.com/od/pronunciation/a/The-German-Alphabet.htm
In general it can, here it can't:
The word haben is either 3rd peson plural, meaning they have or 2nd person singular/plural using the formal speach meaning you have.
Furthermore, you put read as a verb, which simply is the wrong translation of haben:
- haben: to have, own (sth.)
- lesen: to read
The r at the end should be well hearable in einer. I have no comparison right now, but for the sound here it seems ok to me.
Apart from that, consider it as an additional training, since you should be able top distinguish the two by knowing the gender of the noun (Zeitung here) and the case. Although I understand that that might not be so easy for non natives.