This is quite normal for a child of 3.5 years. In their late second to early third year, children enter a stage of wanting to a degree of autonomy or self determination. Of course their choices are limited to what they know, but they very much want to choose. This phase is so common and important that addressing it is built into the 18 month Pediatric visit guidelines by the American Academy of Pediatrics. (See link) And yes, her brother is entering that phase, too, but not because he's observing his sister (well, maybe a little bit.) He's entering it because it's time.
Perhaps I shouldn't let her do it but should instruct her not to fuss for silly reasons.
Please consider taking a positive and proactive approach. As you said, "[S]he's a good kid, pretty clever, listens well, and likes trying to help when adults are doing things." Encourage that spirit! The more she learns she can do, the more helpful she will become (even if it is only in smaller matters, e.g. of self-care.)
After the fact, your daughter realized that she wanted to turn on the faucet herself. Being proactive necessitates that you slow down to anticipate choices your daughter can make. If she needs to wash her hands, tell her, then ask her if she would like to do it herself (with supervision). Let her do what she can, help her with what she can't, then praise the effort (as opposed to the outcome.)*
Try to give her as many choices as you can. "Which of these do you want to wear today?" (If you're adventurous and also very patient, you can ask her instead, "What do you want to wear today," but be forewarned she might choose her Sunday best, or clothing that's inappropriate for the weather. However, these are excellent teaching/learning opportunities, for example, she can't paint in her Sunday best; she will be cold outside in shorts and a tee shirt, and it will further limit the time she spends outside. Eventually she will learn activity0appropriate clothing choices.) "What game would you like to play? Do you want to set up (the board, etc.)?" "Would you like to set the table for your breakfast?"** There are many opportunities for her to choose throughout the day. You don't need to let her make all the choices she can, but the more choices she has, the less frustrated she will be when she doesn't get a choice.
Although the link applies to the school environment, the principles are the same. Self-control decreases frustration and conflict, and builds self-esteem.
*Praising the effort will help her believe that trying and failing is ok and normal. Praising the outcome teaches her that failing is not really ok.
**We put the dishes on a shelf the children could easily reach standing on the floor for just this reason.
Offering Children Choices: Encouraging Autonomy and Learning While Minimizing Conflicts
BRIGHT FUTURES GUIDELINES FOR HEALTH SUPERVISION OF INFANTS, CHILDREN, AND ADOLESCENTS