Can a convention convolution neural network train correctly with different training image size and ratio?
Do not do this.

You are providing the algorithm with examples your training data should conform with the same dimensions otherwise you will skew the weights with alternate examples that do not reflect a true positive as like the original batch. 

Your data preparation is important, if you have the ability to standardise the size then you should do this at the start before training. 

If you have a library of images that you want to train on that are all different sizes; I suggest that you do the following:

Use opencv for the following
1) Write a script that cycles through the images
2) use the script to clip the images on the x:y axis making a square 
     2b)you can automate this by using the click point for centre which then clips and save the new sized image

obviously save all the files in another directory but they should all be standardised now.