From what I’ve seen of R’ Hershel Shechter’s anti-B”D tirades, most of his ta’anos are that Batei Din that allow to’anim are dishonest, which is too general a statement to be accurate. I’ve also seen him spread literal libel about certain Batei Din.
B”D usually charges per hour, as mentioned, and it is split between the plaintiff and defendant. In the extremely rare cases where the plaintiff had no real claim, and the defendant went to b”d only because he didn’t have a choice in the matter, his fee is usually waived. (I don’t know if the plaintiff has to pay it, or if b”d loses out.)
It is wisest to go to a reputable b”d and that neither side should have to’anim. If the dayyanim are talmidei chachaim (if they’re not, find a different b”d), to’anim are unnecessary and are just an additional expense. R’ Shechter also has a valid point that to’anim could drag out a case, and that in inept batei din, the to’anim could cause a misruling. This isn’t c”v saying that all to’anim are crooks, and k”v not saying that all batei din that allow to’anim are corrupt.