Gaia Data Release 3 (DR3) data have revealed new massive runaway stars, while large spectroscopic surveys now enable detailed characterization studies. However, the relative contributions of binary supernova (BSS) and dynamical ejection (DES) scenarios to explain their runaway origin remain poorly constrained, particularly in the Milky Way. We aim to characterize the largest sample of Galactic O-type runaway stars ever investigated through their kinematics, rotation, and binarity with the ultimate objective of shedding light into their potential to identify signatures of their dynamical origins. We use the GOSC-Gaia DR3 catalog of normal and runaway stars, and IACOB spectroscopic information to build a sample with 214 O-type stars with available information about projected rotational velocities (vsini), and a subsample of 168 O-type stars with additional information about their likely single (LS) or single-lined (SB1) spectroscopic binary nature. We also consider an additional sample of 65 double-lined (SB2) spectroscopic binaries. We find that among our sample of Galactic O-type runaways, most (74%) have vsini<200km/s. There are no fast-moving (VPEC_2D_>85km/s) runaways being fast rotators (vsini>=200km/s), except for HD 124 979. Runaways show lower SB1 fractions than normal stars, with no runaway SB1 fast-rotating systems; on average, runaways rotate faster than normal stars; and their runaway fraction is higher among fast rotators (44%) vs. the slow rotators (34%). This is consistent with BSS dominance for fast rotators. We also found that SB2 systems hardly reach runaway velocities with a low runaway fraction (10%). Runaways with VPEC_2D_>60km/s are mostly single and interpreted as DES products, while runaways with VPEC_2D_>85km/s are also interpreted as two-step products, with the binary V479 Sct/LS 5039 a likely example. Finally, three of 12 runaway SB1 systems are high-mass X-ray binaries. Our observational study reveals that Galactic O-type runaways are dominated by slow rotators. The study points to a dominance of BSS among fast-rotating runaways, and of DES and two-step among the high-velocity ones. The observed trends provide valuable constraints on models about runaway origins.