Concerning your follow-up question (iii) [and note that follow-up questions are generally frowned upon in this forum, better to ask a new question] there is the following very nice result: For Birkoff-James orthogonality 
it is easy to find examples where $y\perp x$ but 
$\left\|x\right\|/\left\|x+\alpha y\right\| > 1$ for some real $\alpha$, and so 
natural to investigate the largest such value 
$\left\|x\right\|/\left\|x+\alpha y\right\|$ over $X$. 
In "R. L. Thele, Some results on the radial projection in Banach spaces.
Proc. Amer. Math. Soc., 42(2):484--486", it is it is shown that this quantity
is exactly the Lipshitz constant for the radial projection onto the unit ball
in this norm.