Hi,
This is my first crack at this and I'm not a psychologist, but maybe I could be of use anyway. But be warned that this is just my interpretation.
1. The two cardinals are an interesting symbol. What I've found is that they represent relationships. The two cardinals would be a symbol for you and your girlfriend. The clarity of the tattoo undoubtedly then represents your relationship. The unfinished state of the tattoo would be representative of the fact that your relationship seems to be in limbo. The tattoo getting finished would then represent the two of you moving forward in your relationship and thus committing as a couple. So far, these are all symbols that I'm pretty sure of.
I don't know if it meaningful that the apartment didn't look like the actual apartment, but still felt familiar. It could perhaps represent change? My guess would be that this represents the change you want in the state of your relationship, hence the duality of it still being the same apartment, but looking differently. Especially if you see this apartment as a second home, this would bear more meaning.
The fact that she has a tattoo in the dream that represents your relationship even though she doesn't like tattoos in real life could symbolize her reluctance to be in a relationship.
2. Your mind doesn't delete data, it consolidates it (your brain actually stores all the information you take in, but you can't actively recall most of it since the brain deems most of it as of minor importance). And this consolidation doesn't even happen in the dream state (during REM), but instead during NREM. TL;DR, no you're brain is not deleting any information.
3. Freud thought of dreams as wish fulfillment. But this is only partially true, since our unconscious is also trying to communicate that there is a problem. This however can align as simple wish fulfillment. This is probably the case, but do understand that the dream probably means that you really yearn for a meaningful relationship with this person. It is less something you want and more something you most deeply desire. The fact that you felt at peace when you put her hand on hers and sank into the couch really symbolizes that this is the state your unconscious wants to be in.
People need meaning in their life and starting a family is one of the (if not the) most meaningful things you can do. Simply put, starting a meaningful relationship with her is probably going to actually put you at peace with yourself, as if you've somehow been completed. I say this last part from personal experience.
4. Is your relationship dead? No, definitely not. You are obviously game for a relationship as the dream suggests, but this is a double edged sword. You want to know when the tattoo is getting finished, i.e. you need to move forward and you don't want to be stuck in relationship-limbo much longer. She needs to decide to go either forward or break up. Your unconsciousness is putting its cards on the table: you want this relationship. If she doesn't, this could mean the end of the relationship.
5. I don't believe that dreams predict the future if that's what you're asking. However, this dream does show what you want the future to be: clear as to where you stand in the relationship and you wanting to move forward.
I hope this helped you out and the best of luck with your relationship! I wish you all the best!
