Brownie leader here! Those activities sounds absolutely fab. Axe throwing is similar to archery, and will be done at targets. As an organisation we don't allow shooting at human/animal shaped targets, so they definitely won't be killing the pigeons themselves! I would advise you to give permission for her to do all the activities. She can always say no if she doesn't want to, but if you don't give permission, she can't decide to do it on camp.
As for the Passion for Fashion "badge". This is actually a Go For It, which are activity packs on a HUGE range of different topics that should be chosen, planned and run by their patrol. It is quite normal for them to bring in equipment for these activities; this is one of the skills they are developing by doing these. I highly doubt their leaders are making them do it. There are as many non-"girly" Go For Its as "girly" ones.
It personally really bugs me when people compare Guiding and Scouting. We are two completely separate organisations and have very different rules. All Scouting and Guiding groups are run by volunteers who have to go through a lot of training to run meetings, and even more to be able to take kids away. I would say it takes me longer to plan a residential trip than the time I spend actually on the trip! Plus I often have to take holiday from work to go.
Because it is run by volunteers, groups will vary wildly depending on the skills of the leadership team. It will also vary depending on what the kids want to do. I can't speak for Scouting, but Guiding is very much girl led; 80% of what we do at my Brownies comes from ideas the girls have given us. So this term we've had a movie night and a walk around a National Trust place nearby. We've got a campfire, a Go Ape session, and making sweets for Father's Day coming up after half term.