Here is the list of core plugins and structure which should not be removed or else things will break.
Yii2 Framework (2.0.12) - https://github.com/yiisoft/yii2
Advanced Template - https://github.com/yiisoft/yii2-app-advanced
Password Validator - https://github.com/kartik-v/yii2-password
Domain Validator - https://github.com/dmitry-kulikov/yii2-domain-validator
At Least Validator - https://github.com/code-on-yii/yii2-at-least-validator
Date Time Compare Validator - https://github.com/nepstor/yii2-datetime-compare-validator
Black List Validator - https://github.com/chouaibi/yii2-blacklist-validator
Tree Manager - https://github.com/kartik-v/yii2-tree-manager
Kartik Widgets - https://github.com/kartik-v/yii2-widgets
Kartik Date Range Picker - https://github.com/kartik-v/yii2-date-range
Kartik Sortable Input - https://github.com/kartik-v/yii2-sortable-input
Kartik Editable - https://github.com/kartik-v/yii2-editable
Widget Alert - https://github.com/kartik-v/yii2-widget-alert
Sweet Alert - https://github.com/aryelds/yii2-sweet-alert
jQuery UI - https://github.com/yiisoft/yii2-jui
Imagine - https://github.com/yiisoft/yii2-imagine
Imagick - https://github.com/tpmanc/yii2-imagick
Enhanced Gii - https://github.com/mootensai/yii2-enhanced-gii
Collection - https://github.com/yii2mod/collection
Helpers - https://github.com/yii2mod/yii2-helpers
Kartik Helpers - https://github.com/kartik-v/yii2-helpers
HTTP Client - https://github.com/yiisoft/yii2-httpclient
Scheduling - https://github.com/yii2mod/yii2-scheduling
Disable Submit Buttons - https://github.com/Faryshta/yii2-disable-submit-buttons
Admin LTE Template - https://github.com/almasaeed2010/AdminLTE
Yii2 Facades (Yifa) - https://github.com/foqat/yifa
All of the above modules are already enabled in the config files. If you don't like any of them, simply remove them from composer.json and run a composer update.
Please read the respective documentation of above extensions to know their usage.