(Ito–Michler) p does not divide the order of the degree of any (absolutely) irreducible (ordinary) character of G iff G has a normal, abelian, Sylow p-subgroup.
For any subset π of ρ(G)′, it follows that G has a normal, abelian, Hall π-subgroup, and hence also a Hall π′-subgroup. Flipping the roles of π and π′ gives your first existence claim.
Chapter 14 of Isaacs's Character Theory textbook contains this and further results, which I assume are already known. For instance, the classical:
(Blichfeldt) if G has a faithful (possibly reducible, ordinary) character of degree n, then for any π consisting of primes p > n, G has an abelian, Hall π-subgroup.
Blichfeldt's paper below contains the result that G has an abelian, normal, Hall π-subgroup for π a collection of primes p > (n-1)(2n+1). This was refined to:
(Feit–Thompson) if G has a faithful (possibly reducible, ordinary) character of degree n, then for any π consisting of primes p > 2n+1, G has an abelian, normal, Hall π-subgroup.
The more detailed information alluded to in Isaacs's textbook is likely Winter's 1964 paper below. It gives a complete list of groups with non-normal Sylow p-subgroups for p > (2n+1)/3. A non-abelian but normal Sylow π-subgroup is implied by the following:
(Feit) if G has a a faithful, irreducible (ordinary) character of degree p-2 ≥3, then either the Sylow p-subgroup of G is normal, G is a direct product of SL(2,p) and an abelian group, or G is a direct product of 3.Alt(6) and abelian group.
Now each of these results required faithful characters. If you are OK with that, then Berkovich (MR1189113) has a Thompson like hypothesis and an Ito like conclusion (normal Hall π-subgroup containing the nilpotent residual), but it looks at π(G/Z(χ)) too.
The Ito–Michler result is also valid in some sense over finite fields:
(Manz) Every irreducible, p-modular character degree is coprime to p iff G has a normal Sylow p-subgroup.
(Manz–Wolf) If G is p-solvable and every irreducible, p-modular character degree is coprime to q, then Oq′(G), the subgroup generated by the Sylow q-subgroups, is solvable of q-length at most 2 (π-length roughly measures how far away from having a normal Hall π-subgroup you are).
(Navarro, et al.) has generalized both the Ito and Thompson results to blocks in MR1810119, MR1956546, and MR2159761.
A more recent article by (Matterei) gives a reasonable survey of results. The way he rephrases results may be quite interesting to you. He also gives the Carter–Hawkes (MR1199667) results that continue the ideas of Thompson. A recent paper of Dolfi et. al (MR2469367) derives Ito-like results using the orders of a non-vanishing elements g in G, such that χ(g) ≠ 0 for any any ordinary, irreducible character χ and derives Ito's and Thompson's results as corollaries.
Blichfeldt, H. F. "On the order of linear homogeneous groups."
Trans. AMS. 4 (1903), 387–397. JFM 34.0176.02 JSTOR 1986408
Feit, Walter; Thompson, John G. "Groups which have a faithful representation of degree less than (p-1)/2". Pacific J. Math. 11 (1961), 1257–1262. MR133373 euclid.pjm/1103036911
Winter, David L. "Finite groups having a faithful representation of degree less than (2p+1)/3." Amer. J. Math. 86 (1964), 608–618. MR183788 DOI: 10.2307/2373026
Feit, Walter. "On finite linear groups. II." J. Algebra 30 (1974), 496–506. MR357634 DOI: 10.1016/0021-8693(74)90220-8
Manz, Olaf. "On the modular version of Ito's theorem on character degrees for groups of odd order." Nagoya Math. J. 105 (1987), 121–128. MR881011 euclid.nmj/1118780642
Manz, Olaf; Wolf, Thomas R. "Brauer characters of q′-degree in p-solvable groups."
J. Algebra 115 (1988), no. 1, 75–91. MR937602 DOI: 10.1016/0021-8693(88)90283-9
Mattarei, Sandro. "Retrieving information about a group from its character degrees or from its class sizes." Proc. Amer. Math. Soc. 134 (2006), no. 8, 2189–2195. MR2213690 DOI: 10.1090/S0002-9939-06-08274-8