Welcome to the forums and the ever exciting world of budgie ownership!
You have certainly come to the right place as you will find all you need to know on budgie care here.
Since you are very new to budgies, the information on the several articles and sticky threads located at the top part of each section of the forums will be extremely useful as you become more familiar with the species.
You can start by reading the more in depth info here at the Taming and Bonding section and also check the Diet and Nutrition section.
This may come as a shock to you, but by looking at your Rudy's cere I'm seeing a mainly whitish cere, which indicates that you have a young female budgie.
To answer your questions, since you had Rudy for just 8 days, it's not advisable to let her out yet as most likely she is still not fully acclimated and used to the room she's been in. Most likely, she is still in the process of settling in to her new home/environment.
It would also be best for her to master the step up command, so that you are able to take her out and place her back in the cage without any stress at all.
For now, what's most important is to establish the foundations of trust with your budgie and this is done by spending time close to the cage and getting Rudy increasingly more comfortable around you.
You can use your voice in a calm, positive, reassuring and encouraging way to further connect with her and by watching her body language you will be able to tell whether she is receptive to your advances in terms of training or not.
That beak motion you have described where it seems your budgie is chewing something when in fact she is not is called beak grinding and budgies (as well as other pet birds) do this when they are perfectly content and ease.
They will usually beak grind as they are getting ready to nap or to sleep for the night.
The step up will come naturally as your budgie gets more comfortable and is more trusting of you. You can continue to offer Rudy a tasty treat to encourage her to step onto your hand/finger.
It's best to not force your budgie to perch on your finger because by doing so prematurely and when your budgie is not yet ready, this can have a negative reaction and make your budgie more afraid and less receptive to training. You may also get some biting as a response.
When training, it's important proceed at the pet bird's pace and to respect the safe boundaries set by the bird when it comes to the close interactions we have with them.