Have I correctly used 'compare' and 'comparison' in the given sentences?

I'd simply use life instead of lives.

RommelWe always compare our lives with others’ (lives). We compare our worst (works/thoughts) to their best. We never even realize that we let comparison to rob us of joy.

KhoshtipManWe never even realize that we let comparison to rob us of joy.

Why should I include the infinitive to before 'rob'?

The third sentence is unnatural. However, adding "to" before "rob" would make it clearly ungrammatical!

RommelWhy should I include the infinitive to before 'rob'?

We should not. Let is one the exceptional verbs that doesn't need a full-infinitive. Sorry.

KhoshtipMan, the correction you've made before shows that there's to before rob... I'm just wondering why there's an infinitive. Anyway, I understand you. Apology accepted.

KhoshtipMan You should not. Let is one of the exceptional verbs that don't doesn't need a full infinitive.

RommelKhoshtipMan, the correction you've made before shows that there's to before rob... I'm just wondering why there's an infinitive.

As I said, we/you should not use a full-infinitive for that exceptional verb, let. It needs only a bare infinitive. I don't know how I forgot that clear rule on that word.

Rommelthat don't doesn't need

The subject of the does is let.