Taijutsu generally acts with chakra inside the body, so it is indeed mainly based on buffs, such as you said with raising speed, strength ,etc and pure martial arts moves that are significantly intensified by the use of chakra (like the primary lotus, konoha spinkick, and so on)
However, with a decent description, you should always be able to touch the limits of any specialty (for instance, you should be capable of coating yourself in chakra, expelling pure chakra (like the hyuga or how Guy and Lee do it)
As for how elements interact, you obviousl have a grasp of their strengths and weaknesses, but how they interact with other elements depends on what you put in the descriptions of your jutsu, however, in general: two jutsu of two different elements with no apparant elemental advantage or weakness over one another, will basically clash (though exceptions can be made in case of radius jutsu versus straight line jutsu for example).
Always remember that your description more or less makes or breaks a jutsu