If a downld stops or slows to 0 or if I just want to try again with one that won't connect, I'll select it on the downld list, press the stop button. Then I'll re-select it from the search results & press the downld button. I'll try that several times backed up with a new search. That's worked for me since LW2 a few years back. Yes sometimes you need to slave over LW. Have a look at these: HERE & 'maybe' you might pick some useful points up.

For your main problem of losing connection, I'd suggest you drop back your upload bandwidth to about 70/80% or more if necessary so it doesn't play havoc with your connection & you can browse at the same time. With such a connection I'd also disable Ultrapeer (UP) capabilities. Tools>Options>Speed & tick Disable UP Cap. & press Apply.

Also set your downlds to auto-clear. Close any windows you're not using at that time (eg: Connection, Monitor, & LIbrary Windows.) Go to LW menu View>Show/Hide ...

After opening LW, leave it 10 to 20 mins before doing anything. Give it time to load the incompletes & shared files, & try on its own to connect one by one with each incomplete. For different people this can take different amounts of time (even to load the same no. of files.) I share almost 3k files & it takes 10 mins to load them & incompletes on Mac OSX but takes much, much longer on the same computer using OS 9 for half the amount of files. But I still don't use it for about another 10 mins after that. In fact often I'll just turn it on & leave it & check it an hour later to see what's happening. Some of my incompletes often start by themselves whether that takes 20 mins, 40 or 70 mins. The longer LW is on the more stable it is & easier to use.