Great job on the videos and all your runs! I thought you both looked great. The MOST important thing when starting out trialing is to make sure your dog is still having a blast and wanting to be out there with you doing 'agility' . And I saw that on all the runs.
I would be aware and work on the jumping (knocked bars) but not overly. Some of it seems to be the handler focus and timing issues that experience and more trials help smooth out.
I would be more concerned about the contacts, because when those break down at trials they are almost impossible to fix.
If you teach running contacts, then work with your instructor.
It appeared that some of the time you wanted (and got) a 2on/2off. While other times you were just running on with your dog after he happened to run thru the yellow. It was not clear to me from the video that you were requiring a 2on/2off so I'm thinking your dog may be learning that in class he has to stop (or mom marks and we fix) but at trials it is a WHOOHOO and run on regardless.
For me when I start trialing I over exaggerate my 2on/2off even sometimes counting to 2 only after the girls are in position. I have found it hard for me to stop my darn feet from moving which then tends to make the dogs break and move on.
The other thing that is a huge help for my girls is when they do knock a bar and NQ. Because I instantly calm down from the 'oh my gosh I am at a trial' nerves and switch into training mode. This means generally I just RUN because I am no longer worried about knocked bars
plus we are really training any contacts left on the course! Heck I even still do this with my 8 yr old Bretta. Make sure she hits her 2on/2off and put in a front cross, then front again before releasing. Or leave her and go past the next jump before releasing.
And if you do have a running contact then ignore everything I just posted and keep up the good work, you runs were great :-)