How much involvement do scrub nurses have with surgeries?
First, are you sure the person scrubbed was actually a nurse? There's an entire profession who goes to school to learn the ins and outs of scrubbing for surgery: surgical technologists. Nurses can certainly fill this role, but many facilities (at least in my experience) tend to utilize STs because they were specifically educated (nurses need on the job training and a lot of it) and they're also a lower salary scale. The scrub person typically handles the instruments, needles, and other supplies needed for the surgery. In some facilities they may also be responsible for the cleaning and sterilization of reusable items or "picking cases" where they are responsible for collecting all supplies for the next day's surgeries. Depending on facility policies, they may be able to function in a limited assistant role. In my facility, they may hold retractors and suture skin (if they've taken a suture course and have been signed off by three different surgeons each year), but they absolutely cannot use the electrocautery or fire any staplers. Other facilities may allow more.
Is there ways to go through extra certifications to get more hands on responsibility in an OR?
There are options for this- certified first assistant (tends to be the path STs take) and registered nurse first assistant (obviously requires one to be a nurse).
Do NP's have the ability to assist with major surgeries in the OR?
They can, as can PAs. Also depends on the facility- my guess is that in a facility with surgical residents you'll see fewer NPs or PAs in the OR because the residents fulfill that role. In my facility, they are hired by a surgeon/surgeon group and must go through the credentialing process to be given privileges. They are also responsible for more than just being in the OR- they round on patients, write up H&Ps/consults and progress notes, take care of discharge paperwork (orders/prescriptions), and other duties.
Does this mean I am more cut out for pursuing med school?
That really depends on you. Do you want to be the one actually performing the surgery? You're going to need that MD or DO. Do you want to fulfill the assistant role? Then there are other paths.