Both should be fine.
White shoes with velcro straps are generally acceptable at federal facilities. The white color requirement exists at many camps and low-security institutions specifically to distinguish inmates from staff, and velcro is often preferred over laces for practical reasons. That said, confirm with the specific facility before you surrender, because each institution has its own property rules and what is allowed can vary.
On the inhaler, a documented medical need carries weight. Asthma is a legitimate condition, and facilities are required to provide medical care. When you self-surrender, disclose the asthma immediately and make sure you have documentation from your doctor. Whether they allow you to keep the inhaler on your person or require you to check in with medical for doses depends on the facility's protocol, but either way, they cannot simply ignore a daily medical necessity. Get it on the record from day one.
Arriving prepared and organized makes the intake process smoother. Bring only what is on the approved property list, have your medical documentation in hand, and be straightforward with staff during intake. Self-surrendering already puts you in a better position than most people arriving at a facility.