Idle mode, that's the word I was searching for before, not sleep mode.
Yes new searches can bring up different searches, but I think it's still the best method. Using the 'Find/Get More Results' function will usually help to get results that are missing from previous searches. For LW 4, I think you right-click the search tab to use that tool.
GTK-Gnutella, I think BearShare 5 and I think Phex have the option to save searches between sessions. ShareAza I don't know.
No, I'm simply a person who ended up helping on this forum. I'm just a regular user of the gnutella network. A long time (10-11 year) user of LW. And tried the other programs out of curiosity. I tried Phex at same time I started on the network. Phex is a lot better now than it was as it was not very Mac user friendly at the time for me at least.
Something to remember is LW 4 does not delete the incomplete file if you cancel the download. LW 5 does delete the file. Some LW 4 users would use a technique of canceling their downloads, then going to the Library window and finding the incomplete folder, selecting all the files then clicking resume button I think it was. This would force the incomplete files back into the download window and they would all attempt to find hosts. This technique does have some effect. But is a heavy-handed approach.
If you are not happy with LW 4, then FrostWire 4 is always another option. FW 4's design is almost identical to LW 4. No extra options but a good program all the same. Usually upon first running, FrostWire 4 will resume LW's downloads as it uses the identical type of download file and looks to see if you have LW on the computer during the first run. At least this is how it did work, unless they changed it.
GTK-Gnutella is not a bad program but does not have an easy to use interface like all the other gnutella programs. It's almost a nerdy program if I can be forgiven for saying that. lol

From memory, GTK has both a basic and advanced user option. (This is something I asked the LW devs to do, they liked the idea but it was never added since they had other things on their mind at the time, as you might guess the pressures I am talking about.) GTK seems to be the only client that is still developing at regular times. It has some gnutella implementations that are more advanced than even LW 5 had, though these are within the internal engine, not visible.
Phex is a good program. Not quite as easy to use interface as LW or FW. But it does not take too long to learn the ropes with the program. It has a search file-size filter. Phex usually connects easily and does not seem to have an idle mode. Phex tends to connect to more various clients on the network, though did this more-so in the past, as nowadays there are less people using the lesser known clients. As you probably know, LW tends to mostly connect to other LW clones.