Hello confused mom!
They can safely both accept the Linköping University spot and accept the reserve spots at both Uppsala University and Karolinska Institute (Stockholm). If they are admitted in the second admission phase to an education they ranked higher, they are automatically excluded from educations they ranked lower.
For instance, if they get accepted in Stockholm they automatically lose their place in Linköping - which I imagine is what they would prefer, since Stockholm was given a higher priority in their ranking. If they are admitted to Uppsala, they are excluded from both Stockholm and Linköping. This is important to know, though, in case they have a change of heart and develop a preference for Linköping over the other two options.
As for the chances of getting admitted to Uppsala with reserve spot #14 in the BII quota, my gut feeling is that the odds could be something like 50:50. I should be able to give a more sure answer after the second phase of the admission process.
I wish your kid good luck with the admission and I hope this post helped make the situation more clear for you; if there are any more questions, or if something in my answer could have been more precise in some regard, feel free to ask more questions!
