Several solutions
1.)
First you need to write an action that does the job.
Second you select the images in Bridge, go to Tools/Photoshop/Batch and choose your appropriate action and run it. This way you let PS open images from Bridge and process them individually - more efficient that way compared to opening then all at once into PS to process.
So how to do the action?
You record the tweaking part and then simply record a save as and close at end of action. Now this save will be as you have discovered to a specific place. Two ways around that, simply make it to a folder on same HD as your JPEGs and then drag the images after saving into current JPEG folder, which will take a few seconds. Or open one file at start, double click on the save part of action to rejig it each time to go to folder you want.
Renaming the file is easily done in Br using Batch rename, you can keep current name and add text to it.
2.)
Better still is use image processor - Bridge/Tools/Photoshop/Image Processor, which will open files run an action and save in same place. You will still need Br to batch rename - you can save various renaming presets too, which is handy.
3.)
Alternatively and probably easiest - [if you are doing global develop tweaking] why not open all images in ACR and use a develop preset and save as a tiff?
It's so easy and ACR is so very powerful too - I use ACR/LR presets instead of actions for specific looks a lot now.
Cntrl/Cmd+A selects all images, Cntrl/Cmd+R opens them into ACR, Cntrl/Cmd+A Selects all within ACR and you then click on desired your preset and you're done. You can also save then back into same location as a Tiff with orig name plus your "_GE" too.
Make sure that your Br preferences are set so that ACR will open JPEGs + TIffs