How far - I usually try to pick a spot that I want my bait to be, and I throw to that spot rather than think about how far.
Weight - I normally don't go over 3/4 or 1 oz. of weight unless it's a swift river current that sweeps my bait back to the bank. My dad used to fish for cats in the Missouri river which has a very strong current, and his sinker was the cluster gear out of an old Chevy pickup transmission!
And sometimes he couldn't get it to hold in the current!
Length of rod -- I use the same rod for cats as I do for wipers. An 8 foot surf rod with a big spinning reel and 20# mono line.
For me, that's fairly heavy tackle, but you know how catfishermen hate to lose a fish.
Distance- As far as needed i.e. if I am fishing brush I cast to it. But for a standard muddy bottom I cast anywhere from 100 to 250 feet out.
Weight- Depends on type of fishing. Rivers with heavy current anywhere from 8 to 12 ounces. Lake fishing, I base it on how far I am going out. Typically I use 3 to 8 ounces, more often 6-7, to put me out the 200 or so feet I need to go.
Length- I use everything from 6' to 14'. The short rods are for muscling cats out of brush, the bigger ones to better control fish, bait, and float patterens on the river. I have two 10 footers I use the most from Berkley called Glostiks.
My standard rig is as follows: a 6 oz egg sinker slid up my 50 lb mainline, then a big bead, then a 3/0 swivel, then my 100lb leader. I tie two 6 inch pieces of 100lb line to two 7/0 circle hooks and tie them to my leader. If i use a float I slide the float, a Classy or Wazp, onto my 50lb mainline, slide on a bead and tie on a 3/0 swivel. I then attach a 6 inch piece of 100lb leader and tie my hook on. Change the leader length to 24" if usuing two hooks. I tie a bobber knot above the float and adjust it to set the depth I want to fish at. Most cut bait sinks. If you want it to float you can buy styrofoam beads at Hobby Lobby and you add a few just above the bait on the line until it floats. Cats in the spring perfer floating baits because it mocks what they naturally find which is winterkill shad and carp. During the summer and fall I do better with baits on the bottom. But sometimes I mix it up: I put one bait on the bottom hook on the bottom and on the second hook on that pole I put about 4-5 feet up on the main line. I usually use this rig in conjunction with stink baits.
Well if you are fishing for cats most of the time, or a lot of the time, then I think you are justified in getting specialized tackle that is just for catfishing. In that case you would get some pretty heavy stuff. In my case the catfishing time is more in the minority so that's why I use more general purpose tackle. But I can definitely appreciate the guys who concentrate on the cats, because I came from a line of people who did the same.
I usually fish shallow....especially around here in colorado in the summer, alot of lakes dont give you any other options. usually its no more than 4 feet deep in the peak of summer---some of the larger double digit fish ive caught have been in about 18-24 inches of water. i used to do the same back east...with the exception of winter when the cats would ball up in deeper water in schools.
the shallow water technique isnt the best way to fish all of the time...especially as the water cools. however it is one of the only ways on a lot of front range lakes to utilize some sort of assemblance of key terrain on a lake...many of the lakes i fish have nothing as far as cover, humps, or anything along those lines., so using a "flats" area that has some deeper water near by is often the only choice you have in figuring out where the fish might be. i
it works for me....well, sometimes. this year has keep ka-put (well, for me...) as far as catfish are concerned...but it aint over yet
I use plastic floats by Wazp and Classy. For sizes it varies. Shy bites? Pole float. Wind blowing? Weighted floats. River fishing? I can't even get into that because there isn't enough room to explain what I do. I generally use up to twenty different floats for rier fishing because of all the factors involved.