Facebook – How to Hide Activity from Ticker

You could block the respective user(s).

If I share mutual friends with the user I am blocking, will they be able to see my comments to this mutual friend?

Blocking allows you to prevent most interactions with someone on Facebook. People you block won't be able to find you in searches, view your profile, or contact you with pokes, Wall posts, or personal messages. In addition, they will not be able to see your Wall posts, comments on mutual friends’ Walls, or that you are a member of the same group as them.

However, this might not work in certain situations:

Why am I able to see content added by a user that I've blocked on the site?

Blocking a user will prevent any type of direct interaction between you and that user. However, for technical reasons, there are certain instances when an individual you have blocked from your account may be able to view your comments on the site and vice versa, including comments on a mutual friend's photo. Please be assured that your listing will appear in black text, with no profile picture visible or contact links available.

My understanding of the "Hide this post" functionality is that it will hide that post and all future comments, like etc on that particular post.

If another friend posted the same picture again you would have to hide that post too. Facebook isn't intelligent enough to recognise, what are essentially, duplicate posts.