> Are trills only allophonic in Japanese?

Yes. Not only that, Japanese has only one liquid, i.e. doesn't contrast laterals from rhotics.

> When a consonant followed by a high vowel follows /r/, does it always become a trill?

No. [Okada (1999)][1] says:

> /ɽ/, which corresponds to 'r' in Romanization, is postalveolar in place rather than retroflex and mainly occurs medially. Initially and after /ɴ/, it is typically an affricate with short friction, [d̠ɹ̝̆]. A postalveolar [l̠] is not unusual in all positions. Approximant [ɹ] may occasionally occur in some environments.

[Labrune (2014)][2] says:

> The voiced apico-alveolar tap [ɾ] is generally assumed to be the prototypical realization of the liquid consonant in contemporary Japanese [...] Outside of [ɾ], the following phonetic (social or regional) realizations are widely attested: [l], [ɭ], [r], [rː], [d], [ɽ], [ɮ]. [...]

> The short and long apical trills, [r] and [rː] are socially marked variants, characteristic of the popular speech of males from the Tokyo region. The higher the number of trills, the more socially marked the rhotic will be.

Although she specifies Tokyo, my empirical impression is that it's not confined to a particular region and may be heard from a speaker from anywhere (usually male indeed), though it might not connote vulgarity in the same way it does in Tokyo.
  [1]: https://archive.org/stream/rosettaproject_jpn_phon-2#page/n1/mode/1up
  [2]: https://halshs.archives-ouvertes.fr/file/index/docid/951955/filename/the_phonology_of_japanese_r_pre_final.pdf