I wonder what aspects/ideas/concepts/etc. a science of translation would contain that are not also part of Linguistics? I cannot think of any, if you allow a wide enough definition of the purview of Linguistics. Translation Studies could be seen as a subfield of Linguistics, since it deals with a subset of the overall questions in Linguistics.

Absolutely yes, according to my view. You can derive translational data from linguistics analyses. For example, by generating the N-grams or the most frequent words, or the keywords of two comparable corpora (that are corpora composed of documents dealing with the same topic, but in two languages), you can derive translation-relevant information. In other words: if you generate the most frequent words from both corpora, you are likely to obtain words (in two languages) with similar (or the same) meanings, etc.

At the same time, translation is also a separate field of study (and science).

Yes it can. Linguistics broadly covers the study of languages and their structures, history, phonemes and usage. Translation, in my view, is a subfield of linguistics encompassing two of the tenants of general linguistics: structure (understanding how sentences are constructed between two languages and using that to create the closest translation) and usage (knowing the difference between a 'grammatically correct' sentence while simultaneously being aware of a similar colloquialism or dialectal usage that may better portray an idea).

Of course. Even if translation studies can be viewed as separated field, but it is also can be viewed as abranch of linguistics, especially comparative linguistics. Linguistics is the science which studies a language, and translation is the process of rendernig texts from a language to another.

Absolutely not, in my opinion. Translation is a linguistic phenomenon that requires knowledge of the linguistic mechanisms that make intercomprehension possible, although it is sufficiently important to be studied separately. Similarly, phonetics or syntax can be studied independently in a given language or in general linguistics, but that does not mean they constitute sciences separate from linguistics.

Translation was claimed to be part of applied linguistics as an issue of how a literal or figurative text is being translated from a source language into a target one. Recently, translation has been dealt with as an independent branch of linguistic studies that has its own theories which serve the suggested issues of translation

The translation was part of applied linguistics, but gradually, it developed as an individual subject for study. However, some linguists still consider translation to be the study area of applied linguistics.

Translation deals not only with the languages, but also with literatures and cultures, as we rarely translate just linguistic units without the context. Thus, if we see linguistics as a branch of philology, then the translation studies (or translatology) may currently be considered the part of philology as wel, together with linguistics and literature theory.

Translation is part of the science of Linguistics ... and an art.