Hello Shelley! Welcome to the wonderful world of machine embroidery (ME) and AnnTheGran (ATG).
I did look at your design, and indeed there are intentional gaps in your design. That is part of the whimsical feel of the design.
It is unusual to have so many thread changes with just a few colors, and this one does not seem to have a reason to have them. Some designs have a 'layering' effect so that those color changes are necessary. It appears that there is nothing wrong with your sew out.
Indeed, the software you choose can be a challenge. I know many people like PE-Design as well. It is created by Brother who is a very trusted name in ME.
May I recommend that you read this forum post where you will find a few tips for the beginner. I do highly recommend that you read, ask questions and read as much as you can. You will learn a great deal from others, but ultimately, you must put the needle to the fabric and give it your best.
I am wondering about the 'beeping' you spoke about. There are several arrows on that machine, so I recommend that you check for a selection that shows a Needle with a plus and minus sign. You may have been trying to 'layout' key which is only used at the very beginning of a design. The plus key takes you forward either a single stitch or a single color, and likewise for the minus stitch.
Enjoy your beautiful new craft, it is wonderful and the people at ATG, both employees and the users are the best on the Internet.
Pat, The Avid Embroiderer