If you say "la femme mange du riz" it means "the women is eating the rice" / "the women is eating that rice" ==> Its a specific rice (your rice, the rice that was in the kitchen,...) If you say "la femme mange du riz" , you are talking about an "unknown" rice. You can translate it into "the woman is eating rice". I hope this makes it clearer for you
"Du riz" can mean some rice, just as "du pain" means some bread. Theres a grammatical difference of french versus english in this case where the sentence translates to either "The woman eats some rice" or "the woman eats rice". In french, you simply are required to have "de" or contractions thereof in sentences like this.
Verbs that end (infinitive form) with -er (like manger = to eat) are always conjugated the same way : The radical (here mang-) then terminations : (-e, -es, -e, -ons, -ez, -ent). So in this case Je Mange / Tu manges / Il mange / Nous mangEons (you need to add a E) / Vous mangez / Ils mangent.