Issues with training SSD on own dataset
What you are experiencing is called overfitting and it happens because of your very small dataset.

All the model cares about is performance on the training dataset, so given the opportunity, it will simply attempt to memorize it. This is what happens in you case, you feed a model which contains over 130 Million parameters less than 319 images.

So regarding your questions:


The loss function shows a clear case of overfitting.
On general, it is okay to use a trained model, especially when you only have a small dataset, but in your case, the dataset is too small for any deep-learning model. When I say small dataset, I mean 10k images, not several hundreds. 
You should not train for longer time, once the validation loss stops improving, it is a clear sign to stop. There is even a training technique named "early stopping" which is designed to stop training once the validation loss stops to drop.


You have to understand that currently, your dataset of 300 images, is irrelevant to the world of deep-learning. So if you still want to use it for object detection, you need to revert to more classic computer-vision techniques like using HOG or SIFT features, or even manually engineering the features for your special case.