1. Is it possible to update the Toolset Starter Theme with the current version of jQuery?
No it won't be possible to do this as our Toolset Starter theme has been deprecated for some time now and won't be supported. Any updates would need to come from a 3rd party source.
2. If not possible, what is your recommendation for a starter theme which includes support for Bootstrap 3?
Toolset includes the ability to load bootstrap 3/4 so you will be able to use bootstrap 3 if you have the latest version of our Toolset plugins installed. If you go to Toolset->Settings and scroll to Bootstrap Loading you should be able to select the version of Bootstrap that you want to use.
Could you pass back to your team that stopping support for the Toolset Starter Theme is not great. There will be a lots of websites out there using that theme and it's not a simple job to swap over to a new theme.