@truebaran: For your question about $w_3$, it is a fact that if both $w_1$ and $w_2$ are zero for a bundle, then $w_3$ is also zero. More generally the first nonzero $w_i$ always occurs for $i$ a power of 2. This follows from a formula for the action of Steenrod squares on Stiefel-Whitney classes given in the exercises at the end of Section 8 of Milnor-Stasheff. (Incidentally, there seems to be a typo in their formula: $k-m$ should be $m-k$.)

@truebaran: The natural map $H^2(BO(n);{\mathbb Z}_2)\to H^2(BSO(n);{\mathbb Z}_2)$ is the projection ${\mathbb Z}_2\times {\mathbb Z}_2\to {\mathbb Z}_2$ sending $w_1^2$ to $0$ and nonzero on $w_2$. Thus both $w_2$ and $w_1^2 +w_2$ map nontrivially, so it is not true that a natural class in $H^2$ that is nonzero on the tautological oriented bundle must equal $w_2$ since $w_1^2 +w_2$ also has this property.

The theorem that diffeomorphisms of $M$ can be isotoped to take fibers to fibers is a special case of a theorem of Waldhausen in a two-part paper on graph manifolds in Inventiones 3-4 (1967-68). He worked in the PL category but the proof applies also in the smooth category if one uses Cerf's theorem that $\pi_0{\rm Diff}_+(S^3)=0$. Another classical exposition is in Jaco's Lectures on Three-Manifold Topology (1980), Theorem VI.18, which gives the analogous result for Seifert manifolds with non-empty boundary, leaving the closed case as an exercise (using similar methods).

A postscript to Igor Belegradek's comment: The argument in Proposition A.11 of my book shows that a space dominated by a CW complex of dimension $n$ is homotopy equivalent to a CW complex of dimension $n+1$, where the extra dimension arises from a mapping telescope construction. The argument, which is due to J.H.C.Whitehead if I remember correctly, uses only elementary homotopy theory. Using cohomology and Wall's work one obtains the stronger result that the increase in dimension is not really necessary, at least in dimensions 3 and greater.

@Marvin Jay Greenberg: The Osgood reference Siebenmann gives is W. F. Osgood, “On the transformation of the boundary in the case of conformal mapping”, Bull. Amer. Math. Soc. vol. 9 no. 5 (1903), 233–235. This announces some technical results which Osgood says allow him to prove that, given a Jordan curve C, there is a conformal homeomorphism from the interior of C onto the interior of the unit circle that extends continuously to C. He doesn't seem to say the extension is a homeomorphism from C to the circle. This is three years before Schoenflies' paper in the 1906 Math. Annalen.