Correct, emotions are complex psychosomatic events, of which vedana is but a small component.
For the purposes of liberation, Buddhism differentiates between emotions in their affecting or intoxicating function, and emotions experienced as a result.
The first type are called klesha, they are traditionally defined by example, as want, hate, delusion, and all kinds of combinations of the three. (For the sake of accuracy I should say that not all kleshas are exactly emotional in nature... they can be strong mental concepts skewing or clouding one's interpretative and goal-making functions.)
The second type are various shades of "agreable" and "disagreable" experiences, including of course the cornerstone of Buddhism, duhkha and its false opposite, sukha. Again, I'm mixing all kinds of stuff in the same bucket here - some of these experiences are primarily somatic in nature and so would classify as vedana, and some are mostly mental - I suppose the most accurate way of saying would be that they are different mixes of mental and somatic states.
In Buddhism it is generally understood that emotions are conditioned by mental dispositions such as attachments, aversions, prejudices, assumptions, generalizations etc - which in their order come from past experience. Plus, the everpresent component is one's current attitude, conditioned by one's level of insight or ignorance about the nature of phenomena and mind.
The methods for working with emotions vary from one school to another. Traditional summary from later schools, is that earlier schools like Theravada primarily worked with (undesirable) emotions through either suppressing them or switching attention away to a different topic, -- from this we go to Mahayana schools which are said to work with emotions through underlying mental dispositions - by deconstructing the ego complex, and removing attachments, generalizations, assumptions etc. culminating in realization of Emptiness -- and then on to Vajrayana schools which are said to work with emotions directly in their psychosomatic form as energies, by using visualizations and yoga-inspired psychosomatic exercises. This stereotypical classification is obviously wrong, as in reality all schools use combination of all three approaches, presented in various ways. Indeed, if you think about it from psychological perspective, the three approaches are merely different embodiments of the same underlying principle. Especially as it comes to producing desirable emotional states, the schools show surprising unity of approaches, each presenting in their unique style.
To get to your specific questions,
I wonder which method is most conducive to either processing an existing emotion, or release and allow unfelt (repressed/suppressed) emotions to arise.
There is no single method that works for all people in all circumstances, otherwise Buddhism would be a very simple thing. There are cases when suppressing/avoidance /distraction works best, practiced either momentarily or over a long term - there are cases when experiencing emotion in full, without analyzing/feeding its mental stimulus, works best - and then there are cases where analysis of assumptions&attachments is the most effective.
As for unfelt emotions, I think I can safely say that most schools agree that meditation is a very effective practice to help suppressed emotions come up. Also, in Mahayana and Vajrayana schools teachers use all kinds of scenario-based improvisations to have the apprentice meet their suppressed emotion or assumption face-to-face.
I am particularly curious about the relationship of these meditative techniques towards emotions: (1) breathing meditation (2) visualization (3) body awareness (4) reappraising/re-framing cognitively.
To me, (1) breathing meditation is a great way to connect with the body and get in touch with the types of emotions stored in the body that are so deeply habituated as to be effectively unconscious.
(2) As I said, visualization is an active generation technique that may be used to directly stimulate certain emotions. These are primarily practiced in Tibetan Vajrayana schools. These techniques range from, should I say, abstract psychosomatic visualizations (colors, shapes, letters - imagined over various parts of one's body) - to vivid impersonations that involve imagining oneself as a certain persona or deity. complete with not only costume and appearance but most importantly the thoughts, emotions, and attitude that such character would have experienced.
(3) Body awareness is practiced in all schools, usually on beginner's stages - although the significance of this technique cannot be overemphasized. My various teachers were fond of reminding importance of posture, gait, comfort, speed of movement, grace, and generally good connection with the body.
(4) Reappraising/re-framing is taught in all schools, both in form of traditional frameworks, and custom-tailored teacher/student 1:1 work. Theravada, Mahayana/Zen, and Vajrayana has their own favorite themes and contexts that are too numerous and too deep to review in any details here.
Lastly, I wonder whether it would be more beneficial to focus on feelings (vedana) or emotions themselves as an emphasis.
My personal advice is to treat emotions as two separate phenomena: bodily feelings, and mental dispositions that cause them - and to work with each separately, one at a time. Treating emotion as single unit ("here is what happened and now I feel this") can lead to projecting the emotion onto the world ("I can't live without my GF who made me heartbroken; or I hate her etc.") - so in effect you are creating a realm and inhabit it with your imaginary self, at the same time declaring oneself a helpless victim of situation. Instead, treating emotion as bodily feeling + mental disposition allows one to take on master's role and be in charge of one's mind and emotions.
