I think some children/teens have a very heightened idea of 'fair' - my DS was similar for a few years always trying to work out if his sister had had more spent on her than him. (In fact the opposite was true, because his sister wasn't remotely concerned with money and amount spent).
He grew out of it and is now an extremely kind and generous adult, not at all bothered about who spends what on whom, and is very close to and fond of his sister (again there was a bit of an age gap - though not as big as your 2).
She's a teen and they often are self absorbed and a bit blinkered (as loads have said - its brain development) and she's had a very bog change in her life. And to be honest babies aren't that interesting - the consume huge amounts of time and to a teen give very little in return. When the baby start pottering about and talking and playing more, and probably idolising her big sister, then your DD's attitude may very well change.
So on the surface yes, the quibble over money was silly, there's an obvious difference to adults between a car seat and £100. And likely you won;t be spending £100 on the baby's birthday - baby's don't generally have very expensive presents. Maybe your DD could help choose baby's birthday and Christmas presents and have an input so she can see that in terms of presents for occasions teens generally do better than babies and she can also have the pleasure of seeing baby enjoy gifts she has helped choose.
I think your DP sounds lovely and it great he has stepped in to sort the transport issue. As well as missing out with her friends, it may be that she is uncomfortable with her aunt being unwell and as it is serious she may be fearful, worry about what to say etc, but can't put that into words.
I think we all question whether we have done a good job during the teenage years as that is when selfish behaviour tends to manifest itself. In a way it is good she feels she can openly voice her gripes rather than silently seethe. You know that deep down she is a good caring girl - and those qualities haven't disappeared, just got a bit hidden. They will come back and I think over the next few years you will see changes for the good as she grows up.
Make a point of asking about her plans before you make plans that affect her - her social life will be very important to her at the moment. And try if you can to get some time where the tow of you can go off and do stuff together, so she knows she has her time with you. And even start a daily 'together' time when the baby is in bed - bonding over a TV series, or having a hot chocolate and a chat together.
Good luck 
You aren't a shit Mum and your DD isn't spoilt. Emotions are complex and they sometimes emerge in the guise of selfishness or rudeness