OFDM modulator in LTE
Definitely yes. But after IFFT operation data block is CP extended before transmission.
Each sample of OFDM symbol from the output of IFFT is weighted sum of all QAM/PSK modulated subcarriers from the input of IFFT.
In the MIMO case all spatial-time algorithms are applied in the frequency domain. So in transmitter you should do encoding with QAM subcarriers before IFFT and do the distinct IFFT for each spatial channel created. While in receiver you should do decoding after FFT applied separately to each spatial channel again. By doing this you exploit the fact OFDM is multiplexing of the large number (actually M) of narrow band channels and by appropriate processing (actually FFT/IFFT pair and CP insertion technique) we can treat each subcarrier as narrow band. It is very useful in the case of MIMO because MIMO decoding algorithms are much too simpler when dealing with narrow band (so flat fading channel is the case) and for each subcarrier we can define simple MIMO matrix of weights $[W_{ij}]$. That's why OFDM + MIMO is so popular combination today.