I'm no expert but just wanted to post because I'm the mother of a bright 7 yr old who has recently gone through a phase of being very sensitive about everything and throwing tantrums etc. It's normal!
I think 7 is a difficult age because they have definitely left infanthood at that stage and so long to have more control over their lives, but in reality of course, can't really make many serious choices for themselves. I think this can lead to a lot of frustration and anger.
I think this is a phase tbh (my dd is improving by the day with the odd relapse here and there) and of course, with your dd being the eldest, she probably is a little bit jealous of her younger siblings and the attention you (naturally) have to pay to them. She's definitely getting lots of attention from you though (games after tea + 40 mins every day adds up to quite a lot esp. with f/t job!) so don't feel guilty!
Also at 7yrs, the shine and novelty has slightly worn off school. I think they have a greater understanding of how the week pans out and that school is a permanent factor in their lives and that there are certainly things they HAVE to do regularly and it can be a boring (washing, cleaning teeth, changing for activities, homework, getting stuff ready for next day etc etc). Everything isn't always "tea and cakes"!!
As I say, I'm no expert, but the phrase "life isn't worth living" sounds like something she has overheard rather than something a 7 yr old would say "naturally" and she is using it because it presses your buttons (my dd is an expert at this!!)
Only you can make that judgement call though; do you feel in your gut that there is something seriously wrong?
I don't know how to diagnose depression in children but is she withdrawn, quiet, or have her eating and sleeping habits changed recently?
If she seems generally OK then I would hazard a guess that this is more of a social problem at school about learning how to fit in etc. It's a tough one that they all have to go through and they are very fickle with friendships at this age.
Personally, I would just try and be very calm and reassuring with her, try and persuade your dh to keep his temper, make sure she has enough sleep, and perhaps try and get her interested in an activity she enjoys (in order to meet different friends and build her confidence).
You sound like a lovely, concerned, conscientious mother to me. She's lucky to have you! Just try telling yourself, "this too shall pass!!"
Good luck!