The wordpress.com Like button can only be used by logged in users - that is people with a wordpress.com blog or gravatar. So you have to log in to your blog account first, then go to the blog where you want to click the Like button.
As for commenting problems, I've some info and tips about how to get round this problem in the first part of this page in my own blog, that might help.
As for not being able to contact happiness engineers directly, this might help:
To contact WordPress.com staff, directly, go to this page:
Notice, you won’t see a contact form. That’s because they want you to go through hoops first, but it is there. If the suggestions given on the page are of no use to you, what you need to do is click the ‘I didn’t find the right answer’ button, and then you’ll be presented with a small form. Be aware that that is not the contact form. What you need to do is to type a few words to do with your problem into the space given and then click the ‘I still didn’t find the right answer’ button . Next, you’ll be shown the contact form. Make sure you fill in the fields correctly – particularly make sure that the blog URL (web address) is for the blog you’re enquiring about. If it isn’t, click the little down arrow by the side of the URL that is showing and choose the correct one. If it’s not about your blog, then you’ll have to mention it in the content of your query.
Something to bear in mind is that it might take several days for Staff to get back to you. Also, if you contact them on a weekend you won’t hear from them til after it – Staff on WordPress.com rarely work here on weekends. This is true, also, of the forums.
To help them deal with your issue more efficiently, follow this advice:
and, before using their contact form, particularly read the info pertaining to that.