Hi there,
1. There is a known issue relating to this which we have fixed internally. We have helped you applied a temporary fix by applying the following custom css code into this file: JoomlaFolder/templates/jr_leotian/css/custom.css

.ckit-template {

    display: none !important;

}

However, this temporary fix doesn't complete fix the issue of the conversekit icon not showing in other pages because there are numerous js errors coming from your template(http://take.ms/7Q2gN). Conversekit will not load on the page if you have js errors on the page. You should consult your template provider regarding these errors.

2. You can turn on this setting in Conversekit(http://take.ms/cZ2Ia).
3. & 5. These settings actually inherit from EasySocial conversation settings(http://take.ms/aZ1CW).
4. Unfortunately, this is not possible.
6. Thanks for your input on this. If you want, perhaps you can post a feature request on this.