Characteristic classes over the real projective spaces ${\mathbb{P}}_d({\mathbb{R}})$

Question

Inspired by other users --- this and others, also here: We may explore the characteristic classes of real projective spaces ${\mathbb{P}}_d({\mathbb{R}})$, in some sense, it is related to some sort of "sphere" when we consider the most simple case ${\mathbb{P}}_1({\mathbb{R}}) \simeq S^1$. We however like to know general $d$.

It is still worthwhile to know the basic characteristic class data of them:

1. Stiefel–Whitney class $w_i$: ${\mathbb{P}}_d({\mathbb{R}})$ may be non-orientable and non-spin. Note $$ w_1({\mathbb{P}}_d({\mathbb{R}})) =d+1 \mod 2\neq 0, \quad \text{ if $d=0 \mod 2$} $$ $$ w_2({\mathbb{P}}_d({\mathbb{R}}))=\frac{(d+1)d}{2} \mod 2, \text{ if $d \geq 2$} $$ Thus $w_2({\mathbb{P}}_1({\mathbb{R}}))=0$, and for $d \geq 2$, we have $w_2({\mathbb{P}}_d({\mathbb{R}}))=0$ for $d=3$ or $0$ mod $4$.

More generally, we have for other $w_i$: $$ w({\mathbb{P}}_d({\mathbb{R}}))={(1+a)}^{d+1}, $$ where $H^*({\mathbb{P}}_d({\mathbb{R}}),\mathbb{Z}_2)=\frac{\mathbb{Z}_2[a]}{a^{d+1}}$

2. Euler class: $$ e({\mathbb{P}}_d({\mathbb{R}}))=? $$

3. Wu class $u_i$: is related to the Stiefel–Whitney class $w_i$ through Stenrod square, so $$ u_i({\mathbb{P}}_d({\mathbb{R}}))=? $$

4. Pontryagin class $p_i$: $$ p_i({\mathbb{P}}_d({\mathbb{R}}))=? \in H^{4i}({\mathbb{P}}_d({\mathbb{R}}), \mathbb{Z}) $$ We can consider all the $d=0 \pmod 4$ dimensions. e.g. $p_1(T{\mathbb{P}}_4({\mathbb{R}}))=0$ (yes?) and the frame bundle $p_1(F{\mathbb{P}}_d({\mathbb{R}}))=?$.

5. Are there other helpful / useful Characteristic classes for real projective spaces ${\mathbb{P}}_d({\mathbb{R}})$ that are nontrivial (or trivial)? A list of such info is greatly appreciated.

Edit. Chern class $c_i$ removed.