Yes, you can output Like buttons using the CB Activity module or CB Activity Bot. By default the module Like usage will like the page by its Itemid. You can supply your own asset for both the module and bot usages (e.g. 'fan.movies'). Be sure to read the parameters carefully under the Likes tab for the module and the description of the CB Activity Bot for Joomla content prepare usages for usage information.

It won't log activity for likes of your asset unless you've created a CB Auto Action to do that, but even then the activity display code has no clue what they liked beyond the asset. Best way to do this is first add the system action so it logs the activity then to extend the core activity parsing template file with your own code for it to parse the asset into some sort of display. You can do that in the below template file.

/components/com_comprofiler/plugin/user/plug_cbactivity/templates/default/activity_core.php

You'll want to make a new folder next to the "default" folder and place a copy of activity_core.php into it then select your folder as the template for CB Activity so you don't lose your changes every time you update CB Activity.

What are you expecting the comment to do? Log an activity entry about the comment? Please clarify what exactly you're wanting it to do. There is no parsing for custom assets for activity display. You need to add that as explained above. The system action "CB Activity - Comment" also only logs activity for comments on activity, it does not log activity for any comment as it has no idea what the asset "test.123" for example even means or what it should display.