When I hover over a button and the tooltip appears, the buttons jump. If I disable the tooltip, it does not jump. Additionally, the right button loses the rounded edges. How c
To avoid the jump, you need to set the container attribute. This is documented.
When using tooltips and popovers with the Bootstrap input groups, you'll have to set the container option to avoid unwanted side effects.
Try the following:
$('[rel=tooltip]').tooltip({container: 'body'});