I have now been dropped seven times and this results in saved games.
Regarding beeing dropped my only recommendation is not play players with a negative reputation. Though there may be few players who have a negative reputation while not deservig it, it's still the most relyable information.
Secondly, I am embarrassed to say that I do not know how to recall a saved game where a bad connection has caused a drop. I cannot find any reference in the manual to this.
If you use JavaFIBS, you can invite a player by just opening the invite menu. Select a player in the players list, then use the pull down menu:

Then push the buttom to the right and the player is invited to resume the match (red arrow).
Finally, I still cannot make the programmable buttons work in JavaFIBS. I am trying to programme 'tell RepBot ask %player'. I does not do anything. Also, can I add the button (once it works) to the menu bar?
'RepBot ask %Player' only works for players in the players list. You have to select a player there and then it works. To do the same for inviters, you need the command 'RepBot ask %Inviter' and to have selected the inviter in the invite window. Btw, in both cases, it has to be a
capital first letter (i.e. %Player instead of %player!).
Getting the programmable buttons work is sometimes a bit tricky. I therefore prefer to edit the respective configuration file (called "button") directly. It's found in the directory JavaFIBS2001/user/name_of_user/. My "buttons" file i.e. looks like that:
kibitz :-)
images\happysmiley.gif
kibitz ;-)
images\winkingsmiley.gif
kibitz :-/
images\slashsmiley.gif
kibitz :-(
images\angrysmiley.gif
toggle ready
images\pready.gif
set sortwho rating
images\say1.gif
who
images\say2.gif
set sortwho rrating
images\say3.gif
who
images\say4.gif
set sortwho name
images\say5.gif
who
images\say6.gif
k Hello & gl from Germany!
images\greet.gif
tell RepBot ask %Inviter
images\sayA.gif
tell RepBot list %Inviter
images\sayL.gif
tell RepBot ask %Player
images\punknown.gif
tell RepBot list %Player
images\doppeltesfragezeichen.gif
away I am away, back in a few minutes (probably).
images\away.gif
back
images\sayB.gif
The "buttons" file can be edited with every normal text editor. Make sure, the file is saved with the same name as before, so that no ending .txt is added.
Hope that helped,
Hardy

P.S.: And don't forget to visit
www.hardys-backgammon-pages.com 
At the section "Articles" you find some hints about JavaFIBS as well.