Are you using one of the Premiere Pro PIP presets? Embarassingly I did not even know they existed ...
But now I've had a play with them ...
if you are using a PIP there are two ways I can think of to add a feather to the edge of your PIP.
First is add the crop filter to your PIP layer and crop the edges slightly using the Left, Top, Right and Bottom slider and play around with the Edge Feather slider to taste.
Second is to add a 'mask' to the PIP layer. Double click the PIP layer and then look in the effects control panel and apply an opacity mask. (click the square icon under 'opacity'). You'll then be able to adjust the size of the mask by moving the mask outline that will have appeared in the 'program' monitor. By default the mask will probably be too small - so drag out the corners with your mouse. Feathering for a mask is also under Opacity > Mask > Mask Feather in the Effects Control panel.
To adjust the speed - also in the effects control panel up the top under 'Motion' you'll see Position and Scale. Across from these in the effect control panel 'timeline' to the right you'll see diamond shaped icons or 'keyframes'. If you can't see this part of the Effects Contol panel - click the small dotted triangle in the top right hand corner of this panel to open it out to the right.
These keyframes control the speed of both position and scale. Move them closer to each other to make a position (or scale) happen faster.
Let us know if that helps.