My website has errors in the source code, and there's nothing I can do

Question

Hello! So, a summary of my issue. Basically, I use Network Solutions as my website hosting service. I've been coming across issues with my images not showing up on google, and I've been looking at everything to try and figure out why. Here's what I did to find out:
I uploaded a test product to my website. Here is the URL:
https://compressedairpartscompany.com/35148-86722waterbottleoem.aspx
After inspecting this URL in google's structured data tester, I saw that it returned 2 errors and a bunch of warnings for things like type, id, name, brand, description and image. All of this information is in my website. So I was very confused when I stumbled upon this. 
I did some further digging and learned what structured data is and how to code it. So I uploaded another test part to my website: https://compressedairpartscompany.com/shh12-cm357coffeemugoem.aspx
Inspecting this one reveals the same errors as were on the other product, however, I also added a segment of code to the Head<HTML> editor on that specific product page. You can see the almost completed segment of structured data I added MYSELF, and it returns no errors. However, since there are still errors in my source code, Google must see that and think, we don't want these guys showing up for results because their website has errors in it. Which makes sense.
So, I contacted my website hosting service and informed them of the issue and asked if there was any way for me to correct that and remove the errors so we could become eligible for rich results and higher search rankings. He basically said that one of the drawbacks of using their hosting service is that the source code is not accessible in any way, meaning there is no way for me to change the error that is being generated by their system. After doing some further research, it seems as though EVERY WEBSITE using Network Solutions has the same errors in their source code/structured data as I do. Doing a quick google search for some of my competitor's product images that are also using network solutions yields similar results. This makes me think that we are kind of screwed. SO... I have 2 questions considering all of that information.
1. Are these errors affecting my search performance? Is it considered "fatal" to have these errors, so much so that google wont even display our images? (Maybe obvious but I want to hear from the experts)
2. What should we do? Would it be a good idea to purchase a new domain from a different web hosting service? Would it be more beneficial or less considering the new age of the domain, how long we have been using the same website... We have over 100,000 products that we have listed for sale on our website and I just don't know if I am confident in having a web hosting service that generates errors in their code and doesn't provide a way for the customer to fix it. Especially if this could be a big issue. Any suggestions for a more polished, more industrial hosting service that would better suit our business' needs?
This would be a big move for our company, and I just want to make sure this would be the right thing to do in this scenario. Thanks so much for your time and any responses, I really appreciate it.