Usually when I find my marksdwarves refuse to train with archery targets, its because the direction is not right.
to be thorough:
make a target
make a room from the target
pick a direction (using "awds" it should say "shooting from left to right" or something) <-- this step is my #1 cause for refusal to train
then - / + to the squad you want to train there and "t". Should show a light blue T next to their squad's name
If you know how to setup melee dwarves
to spar (and if you have decent hammerers then I'm sure you do) then you can ignore this bit.
Make sure their schedule/ alert has them training. Notice that the alerts screen and the schedule screen are connected and schedules default to the active/training alert. (So you can setup separate schedules and switch between them at the flick of a few buttons) So unless you've messed with things a bit (then I assume you know what you're doing) then you want them on Active/training.
If they stand around the meeting hall as recruits / xbowdwarves then there's something wrong with your ammo supply or your training room (the Archery target). If they don't turn into recruits / xbowdwarves (act like civs) then there is something wrong with your schedules/ alerts (they are inactive, or not scheduled)
Do they shoot bolts in combat or bash? If they bash, then check that they have bolts:
"m" "e" "P" and check to make sure they each have quivers as well as crossbows. Then "v"eiw their "i"nventory, -/+ to their quiver and check its contents for bolts. but if they were firing at loose gobs, then I'm sure you have that right. You could confirm that they were firing with the "r"eport screen. and an alternative to a shooting gallery would be to send them after wild animals (I recommend harmless animals at first so they don't try to bash a cave croc's head in with a bone crossbow or something.)
If they don't have bolts then they may just be behaving a little glitchy:
This seems to happen to ppl occasionally and I don't know a definite fix. try disbanding/ reinstating the squad, flooding the fort with extra bolts (hunter's or other squads may have already claimed your entire supply, even if they don't carry them around) assigning 1 stack for training and combat (make it bigger if you have lots of dwarves in the squad -- Maybe, 2 stacks of 25 for each dwarf. so 100 for 2, 500 for 10) giving move orders for a moment (to jar them awake), checking "z" stocks for forbidden equipment or shortages or just flood the fort with magma ("see what you made me do!?")
As for the interruptions, the only thing I can think of is that somehow civs DO somehow have line of sight. Fortifications are totally see through, some glass products may be see through (portals?), grates and z levels don't block LoS either. you could drop a bow gob in and see if he shoots at any civs...