The best thing to do is to have your cooling system checked to find the cause of the broken tube. The system may have other issues that would need addressed at the same time. That being the case, I recommend...
I would agree with that mechanic, but it could also be low on coolant. I would check the level make sure its full, and then start it up, have a certified mechanic such as one from YourMechanic, feel the upper...
For this type of work the cost will vary tremendously depending on where you have it fixed. Parts prices and hourly rates will vary depending on the shop. The one thing that does not change is the time it takes...
There can be a lot of things that could cause all of your warning lights (https://www.yourmechanic.com/article/what-does-it-mean-when-the-dashboard-warning-lights-blink) to come on. Low battery voltage loves to reek havoc on any electrical systems and a low charging alternator would be the simplest repair....