For models $M_1,M_2$ with parameters $\theta_1,\theta_2$ and priors $\pi_j$ , the Bayes factor is the ratio of marginal (integrated) likelihoods:

$BF_{12}=\frac{m_1(\uline{x})}{m_2(\uline{x})}, \qquad m_j(\uline{x})=\int f_j(\uline{x}\mid\theta_j)\,\pi_j(\theta_j)\,d\theta_j .$

(I) is true — the marginal likelihood is precisely what is being ratioed; the parameters are integrated out rather than maximised over.

(II) is true. Since

$\underbrace{\frac{\mathbb{P}(M_1\mid \uline{x})}{\mathbb{P}(M_2\mid \uline{x})}}_{\text{posterior odds}} =BF_{12}\times \underbrace{\frac{\mathbb{P}(M_1)}{\mathbb{P}(M_2)}}_{\text{prior odds}},$

the Bayes factor is exactly the factor by which the data update the odds — the weight of evidence the data supply for one model over the other.

(III) is false. A conditional likelihood is a different device (used, for instance, to eliminate nuisance parameters by conditioning on a sufficient statistic, as in conditional logistic regression). The Bayes factor integrates against the prior; it does not condition.