b and v are pronounced same but d and v are not. Listening to the recording I can see how you heard that though, she is tricky to understand sometimes, though from previous comments it seems she's only hard to understand if your ear is not used to listening for these sounds (native Spanish speakers do not have a problem with the recording).
I took some spanish in high school but forgot it all, except little things like this apparently! Now that I'm hanging out with some Latins, I'm trying to pick it up again, and this is one that I"ve noticed. Another sound I"ve noticed is "y" and "j" they like to reverse those sounds...So Yellow is pronounced "Jello" but Joe is "Yo" for some reason!
YouTube some spanish kid's cartoons, or songs, or check on craiglist or in coffee shops if there is a language exchange happening in your town/city. Around me, there are always people trying to learn english and they'll happily help talk to you in their language, so you both can practice.
The Spanish 'd' can sound hard or soft. At the beginning of a word it is usually hard, and pronounced like the d in day, except the tip of the tongue touches the back of the teeth and not just the ridge of the palate. In the middle of a word it sounds more like a very thick voiced th. Because the speaker is saying "mi día" you run the words together a little bit, just like you do in English. You use the soft d, as if it were in the middle of a word. Here's a good guide to Spanish pronunciation of the letter 'd': http://www.studyspanish.com/pronunciation/letter_d.htm
However, it is MUCH more important to get the vowel sounds correct than to worry about the consonants. Pure vowel sounds will make you easier to understand and give you a more attractive accent.
I am going to differ from the others. In my opinion it is because you have double nouns. "IT" is a pronoun that either stands alone as a generalization as in "it is sunny outside" or as a reference to a previously mentioned noun as in "The ball is red. It is also round." In the second part the it refers to the ball. If you wrote out your sentence in long form with the it replaced with your reference noun, it would read "Today, today is not my day". In the correct answer the it is not there because today is the noun that determines the tense of ser, es. I hope that makes sense.
Kathallia, That would have a completely different meaning. Today it's not my day would be used maybe if it was not your day to do the dishes, Today is not my day, really means that you are having a very bad day, perhaps you dropped all the dishes,then cut your finger picking up the pieces, then your dog bit you! Today is not your day!