Can I Save Open Dialog Windows Size and Filesystem Location?
That functionality is how the Finder works at present. Sizes made to a dialog in an app should be saved across open actions (but the window position isn't saved as this window gets drawn to be centered and a bit below the active window.) Changes to the underlying window are supposed to affect the open dialog.

Are you using safari or some other program to "browse"?

Have you looked at software like DefaultFolderX to add better tracking of the recent folders? I don't know if it will help with the sizing, but it has been doing the default folder trick for ages and ages and is very stable and well supported. 

Lastly, do check how the Apple apps like Safari, TextEdit and Pages work. Many third party applications don't implement file open / file save / file choose according to the user interface guidelines. Much of this is fairly technical but documented at developer.apple.com in the Human Interface Guidelines. See the open and choose discussion here if you want to know how this happens in code.

In a nutshell - OSX does save folder and size while a program is running but not generally saved after the speficic application is quit. A programmer can make their program do anything - so even if OSX wants to go to the last folder, if the program says to always open the home folder you will need a plug in like DefaultFolder or that program to change.