Is it accurate to claim that the underlying level is (only) characterizable by a phonemic representaiton of a word and the surface level is (only) characterizable by a phonetic representation of a word?
That is one theory of underlying forms and phonemes, which depends on how you define "phoneme". If you define phoneme as "anything that can appear in underlying forms", and if you likewise define "surface level" and "phonetic representation" in an analogously tautological manner, then that would be true. There are, however, other theories especially hinging on what a "phoneme" is.
The classical taxonomic phoneme is essentially a partitioning of surface forms, which unifies non-contrastive variants (e.g. English [t, ʔ, tʰ, ɾ]) being grouped into a single concept, the /t/ phoneme). Accordingly, allophones are not phonemes, and by the above reasoning you would expect that allophones are not present in underlying forms. That is one position, but the alternative position is that underlying forms can contain allophones, especially in no-alternating cases. Under the "allophones can be underlying" view, the underlying form of "water" could be /waɾɹ̩/, and an underlying form would not be composed solely of phonemes. There has been a tradition of gratuitously requiring underlying forms to be abstract to the point of never containing "allophones", but there really haven't been any good arguments for that requirement (which basically follows from making "economy of inventory" be an inviolable principle of analysis). The alternative, that allophones can be underlyingly present, has the advantage of being consistent with the well-known fact that allophonic differences can become phonemicized.
In addition, underlying representations, at least under contemporary "non-segmental" theories (anything even mildly autosegmental) contain things that are not phonemes – e.g. floating tones, empty consonants. Of course if you simply define "phoneme" as "anything that can appear in underlying form", floating tones and empty C's would be phonemes. So it really depends on how you define "phoneme".
Regarding the "surface level" question, there is probably less theory-dependency, as long as you distinguish "surface level" from "phonetic output" (where "surface level" would be the last stage in a phonological derivation – necessarily a representational entity, and "phonetic output" would be physically-implemented facts such as patterns of articulation and acoustic outputs).
Yes, even when you have a perfect one-to-one correspondence between the phonemes and (allo)phones used: even when the language has literally no phonetic variation/allophony, [ɡeɡe] will always be the phonetic/surface equivalent of /ɡeɡe/. This is a strictly theoretical approach, clearly.
Addendum: there are even deeper levels of analysis: archiphonemes and morphophonemes are sometimes mentioned in such contexts, and they tend to (in the roughest possible of terms) act in a manner analogous to how phonemes relate to phones.