You should definitely attend school events, extra-curricular performances, sports meets, birthday parties and ANY kind of celebration of your SD and her accomplishments, even if it makes Mom uncomfortable. That's how you show that you're interested in your SD, you support her and you support your husband being an involved father. Bringing you is how your husband shows that his relationship with you (and thus, his daughter's connection to you) is serious and permanent: You're not just some mistress he wants to keep hidden, to keep from making waves with the mother of his children; you are his wife and you're here to stay.
But be gracious about Mom's insecurity over you. If she attacks you - publicly or privately - for attending SD's events, respond quietly and nonconfrontationally. Do not promise to stay away in future, but you might say, "The last thing I wanted was to offend you. Everyone knows you're an amazing Mom and I don't want to take your place at all. I just wanted to be here to see SD's performance and be one of the people clapping for her. I hope you'll try to see that I'm not trying to hurt you, by coming." If you give in to an impulse to tell her off, because she was rude to you first, you will only confirm her fears that you're out to replace her with her kids, just like you have, with her ex-husband. Let it be clear that those fears are irrational and she may get over them, in time.
If SD has two parents who are both fully involved, then she doesn't need a third parent, in terms of volunteering at school, attending parent-teacher conferences, doctors' appts., etc. Those things should be primarily about the kid. If you doing them shifts the focus to Mom's resentment of you, then there's no benefit in you participating. However:
>> You are your DH's partner, so if something came up that prevented him from attending a teacher conference, IMO it would be better if you went in his place, than for neither of you to go and only the Mom, especially if she isn't great about communicating with your DH in detail about their kid. The most important thing is that your DH stay aware of what's going on with his kid at school. The Mom's hostility toward you IS a consideration, but it's a secondary one.
>> If Mom becomes less hostile toward you in the future, there's no reason you can't go to these things. But it's not necessary. For example, I go to pretty much everything, for my DSS and volunteer at his school and activities. But he lives with us and his Mom's across the country. So, like it or not, I'm doing a fair amount of the raising of him, not just step-parenting. But if his Mom were here, and involved, I would probably back off.
In your home, let the relationship evolve naturally. It's a tightrope. You want to be OPEN to doing anything for your SD that you'd do for your biological children, because if you and your DH have children together in the future, you don't want SD to feel like a second-class citizen in your home. Yet, you don't want to make her uncomfortable - or overshadow your DH, as a parent - by insisting on doing things for her, if your DH is willing/able, or she'd rather have him do them. Just take each situation as it comes and try to read your SD's cues. You may need to just be a friend, a fun aunt, for now; and it may evolve into more. For example, when DSS first moved in with us, he would only hang out in DH's and my bed (not sleeping together, but flopping down together on top of the covers to watch a movie, or talk) if DH was there. Now, if he passes by our room and the door is open and the 3-year-old and I are watching a Disney movie together; or he wants to talk to me about something, he'll just hop on the bed and hang out. But that wasn't something I could have planned. If I had pointedly invited him to join in the family lounging, before he felt comfortable doing it on his own, it would have been awkward. It just had to evolve.