The Haar basis is unconditional in such spaces iff the so-called Boyd indices are non-trivial (i.e, the lower index is strictly greater than 1 and the upper index is finite). The situation is very much like in $L_p$ spaces. See Lindenstrauss-Tzafriri the second volume, Theorem 2.c.6. You can find the relevant background in the same section.

You can't have an unconditional basis for $L_p$ which is orthonormal in $L_2$ unless $p=2$. The trigonometric system is a Schauder basis for such spaces again in the case when the Boyd indices are non-trivial, see 2.c.16 of the same book mentioned above.

The Haar basis is unconditional in such spaces iff the so-called Boyd indices are non-trivial (i.e, the lower index is strictly greater than 1 and the upper index is finite). The situation is very much like in $L_p$ spaces. See Lindenstrauss-Tzafriri the second volume, Theorem 2.c.6. You can find the relevant background in the same section.

You can't have a semi-normalized unconditional basis for $L_p$ that is orthonormal in $L_2$ unless $p=2$, but the Haar basis is an unconditional orthogonal basis for $L_p$, $1<p<\infty$. The trigonometric system is a Schauder basis for such spaces again in the case when the Boyd indices are non-trivial, see 2.c.16 of the same book mentioned above.

The Haar basis is unconditional in such spaces iff the so-called Boyd indices are non-trivial (i.e, the lower index is strictly greater than 1 and the upper index is finite). The situation is very much like in $L_p$ spaces. See Lindenstrauss-Tzafriri the second volume, Theorem 2.c.6. You can find the relevant background in the same section.

You can't have an unconditional basis for $L_p$ which is orthonormal in $L_2$ unless $p=2$. I don't know if there are such conditional bases for these spaces though.

The Haar basis is unconditional in such spaces iff the so-called Boyd indices are non-trivial (i.e, the lower index is strictly greater than 1 and the upper index is finite). The situation is very much like in $L_p$ spaces. See Lindenstrauss-Tzafriri the second volume, Theorem 2.c.6. You can find the relevant background in the same section.

You can't have an unconditional basis for $L_p$ which is orthonormal in $L_2$ unless $p=2$. The trigonometric system is a Schauder basis for such spaces again in the case when the Boyd indices are non-trivial, see 2.c.16 of the same book mentioned above.