It's really hard at first. You definitely need to persevere with the carseat. You can't be trapped for a year. How is she in a sling? You could opt for that instead of a stroller. And google kids songs and relearn them. Or put on a kids station on pandora for a refresher course. Dd1s faves are twinkle twinkle, old macdonald, bingo, itsy bitst bspider and hands down all time favorite, put it on repeat is baby beluga.
It's a bummer she is fussy alot, hopefully she outgrows that. Just keep doing a little bit until she cries then soothe her and do a few other things.
And for the play mat, we had a simple one bar which dd destroyed and I'm not getting a new one. But, we hung extra toys from it with those linky rings. Carseat toys are good options to hang on those bars. I wouldn't buy a new one since it will only last a few months, but that's just me being cheap.