1. One paper which does which addresses your first question (by giving an example) is

Galkin, Sergey; Katzarkov, Ludmil; Mellit, Anton; Shinder, Evgeny, Derived categories of Keum's fake projective planes, Adv. Math. 278, 238-253 (2015). ZBL1327.14081.

There are other examples, https://arxiv.org/abs/1410.3098v1 lists some in the introduction.

2. Your second question is an open question I think, except that maybe we know that this is not possible in easy cases (such as $\mathbb{P}^2$, for which transitivity of the braid group action is known).

3. Your third question is the holy grail in the construction of exceptional collections, and no such thing is known as far as I know.

1. One paper which does (edit: not, I missed the word rational) address your first question (by giving an example) is

Galkin, Sergey; Katzarkov, Ludmil; Mellit, Anton; Shinder, Evgeny, Derived categories of Keum's fake projective planes, Adv. Math. 278, 238-253 (2015). ZBL1327.14081.

There are other examples, https://arxiv.org/abs/1410.3098v1 lists some in the introduction.

2. Your second question is an open question I think, except that maybe we know that this is not possible in easy cases (such as $\mathbb{P}^2$, for which transitivity of the braid group action is known).

3. Your third question is the holy grail in the construction of exceptional collections, and no such thing is known as far as I know.