To realize orthogonal frequency division multiplexing (OFDM)-based grant-free access for wideband systems under frequency-selective fading, existing device activity detection and channel estimation methods need substantial accuracy improvement or computation time reduction. In this paper, we aim to resolve this issue. First, we present an exact time-domain signal model for OFDM-based grant-free access under frequency-selective fading. Then, we present a maximum a posteriori (MAP)-based device activity detection problem and two minimum mean square error (MMSE)-based channel estimation problems. The MAP-based device activity detection problem and one of the MMSE-based channel estimation problems are formulated for the first time. Next, we build a new factor graph that captures the exact statistics of time-domain channels and device activities. Based on it, we propose two approximate message passing (AMP)-based algorithms, AMP-A-EC and AMP-A-AC, to approximately solve the MAP-based device activity detection problem and two MMSE-based channel estimation problems. Both proposed algorithms alleviate the AMP's inherent convergence problem when the pilot length is smaller or comparable to the number of active devices. Then, we analyze AMP-A-EC's error probability of activity detection and mean square error (MSE) of channel estimation via state evolution and show that AMP-A-AC has the lower computational complexity (in dominant term). Finally, numerical results show the two proposed AMP-based algorithms' superior performance and respective preferable regions, revealing their significant values for OFDM-based grant-free access.