Encoder-Decoder LSTM for Trajectory Prediction
There are multiple questions in your description:

The input to LSTMs are normally continuous representations. In NLP, you normally embed discrete elements as vectors in a continuous representation space and then you pass these vectors to an LSTM. You already have continuous representations, so you just pass your 2-dimensional vectors as input to the LSTM.
In NLP neural networks, like most neural networks, the training happens passing as input not one sample, but a minibatch of N samples. This N most of the times is chosen as the maximum number that makes the model, data and intermediate computations fit in the GPU memory.
The 3D array expected as input by the LSTM has shape [N, timesteps, feature]. In your case, this would be [N, 125, 2].
I think you don't need an encoder-decoder architecture, only the encoder. Therefore, a single LSTM would suffice. You would train it to receive any number of input elements and predict the next one. If you want more predictions ahead, you can feed the model's own predictions as input, autoregressively. To find an analogy in the NLP world, your model would be a language model, which receives words (or letters) and generate the following word.

P.S.: I may be wrong about you needing only the encoder instead of an encoder-decoder architecture, as I don't know the specific nature of the predictions you want to make.