I'm not sure about the windows version, but the mac version remembers which was the previous search type. But not each time you open LW. On the mac, if you Control+Click onto the top of the column bar you get many more options for sorting & columns to display. I don't know what the equivalent keyboard key to use for this is on windows LW. ALso do the same click on other parts of LW & you find more options in drop down window choices.

Research button: if you utilise magnetmix & search bitzi you can find out about files.

So basically, all the things you request are already within LW. You need to investigate a bit more to find them.