The rule is a bit of a theoretical anomaly in mixing SPE formalism with apparent autosegmental assumptions (features being "activated" with implications that features may be underspecified, plus autosegmental ~ feature-geometric features), so you need to accompany the rule with a summary of the phonological theory that you are assuming. In the context of SPE rule theory, what your rule says, applied to Turkish, is that all lingual and labial consonants, and the vowels /e ø/, become lax in a context. If you want to specifically restrict this to /e/, you have to add more features to say "non-round vowels".
The context says (roughly) "before {n,l,r,j,m} within the same syllable", but the angled brackets complicates the rule. The formal definition of angled brackets in SPE is a bit of a mess, but the standard interpretation of the notation is that "A → B /[a,<y>] __<z> Q" means that "A → B /[a,+y] __z Q" and "A → B /[a,–y] __ Q". This means that z can intervene between A and Q if the preceding segment is featurally [a,+y] (whatever a is, and it must be specifically plus y), or else if A is immediately before Q and is preceded by a segment that is [a,–y]. The effect of putting [+tap] in angled brackets is to say that {n,l,r,j,m} trigger this laxing rule very generally, but r triggers it only in the specific context that the vowel is word-initial (for example erdogan and not pervez). I cannot tell if that is what you wanted to say. In general, the correct way to write a rule is to say in plain language what you want the rule to do, then find the matching formal expression. I can't figure out what you say the rule is supposed to do.
Your first sentence suggests that you did mean this, so I'd independently like to see proof that this is a factually correct description of the language (a personal curiosity thing). Your concern with "redundant" features is not without merit, but in a vacuum it can't be addressed. Who cares if a feature is redundant? (I do, actually, though technically I'd say the problem is with unmotivated features). My theory of features evaluates features "as they exist in a grammar", so if X is unnecessary, it is unnecessary, and does not have a special status depending on whether some other features are assigned in the derivation. You apparently have a different theory of features. A satisfying answer to your question requires that you set forth your theory of features, a bit much for here, but I'm sympathetic to the problem.
It is not clear what the facts of the language are, but you seem to assume that there is a single rule that derives [ɛ]. The single-rule assumption is often made in phonological analysis, so we don't say that there are is a rule in German devoicing final /g/, and a separate rule devoicing final /b/, and a third one devoicing /d/. However, we also don't say that there is a single rule of vowel harmony in Mongolian, or a single rule of vowel harmony in Turkish, or a single rule of gradation in Finnish, because in the context of "the" theory, it's impossible to state these processes as a single rule. Any theory limits what can be expressed as a single rule: except, it is not clear what constitutes a formally unstateable mapping in SPE theory. An alternative that you need to consider, if there is good evidence that /r/ behaves differently from other consonants which still do the same thing, is that there are two rules.
