yikes you have a lot of pipe length there.
different options for ya....
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OPTION 1. filter + corrgated pipe, + (optional clean out TEEs/WYE's)
filter....
get say a 30 gallon drum or heavy duty trash can (not cheap thin plastic trash can). and burry it up near the shed. and run your downspots / gutters into it.
you can use "toilet flanges" as a cheap bulkhead, to connect the 4" long length of pipe to drum/heavy duty trash can.
for a screen, take sch 40 PVC pipe 4". and set table saw blade, so you cut almost half way through, and make multi cuts 1/4" to 1/2" between each cut. roll pipe over and do other side. this will be like a "well screen" or foot valve screen. and just attaches to the toliet flange inside the drum/trash can.
you have long enough pipe run. that installing a clean out tees at the (90) at least, and perhaps a couple places in the 600 feet run. would be advised, for just in case scenrio of a clog happening. other words if a clog happens, and you can not remove clog, you may have justed wasted a bunch of case.
if you come out the bottom or on side near bottom, it should place ya 2 to 3 feet below ground. keeping ya from needing 6" pipe for that initial 100 foot run.
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OPTION 2.
sch 40 PVC pipe with BELL ENDs, and installing clean outs every 100 foot or less.
corrugated pipe, has all them ribs / bumps inside, that love to catch sticks and leaves. and a clog will happen. PVC pipe has a nice smooth inside surface. that will more likely allow twigs, leaves, etc... to just flow right through and on out of the pipe, having clean outs every 100 feet should allow you to get something down the pipe and clean out the clogs. if something does happen.
the BELL ENDS, vs using a cheap "coupler" fitting. and placing the BELL ENDS in correct direction should also reduce chance of any sort of clog happening
you may want to go down an initial 3 feet near shed. and then once ground starts sloping downwards. allowing pipe to come more near surface of the ground if you want. that way if there is a clog near shed. there will be more water build up within the pipe, and in that (weight of water) to push through the clog or push the clog out. this should also get you away from needing 6" pipe for that initial 100 foot.
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clean outs, are just TEEs or WYE fittings. with a pipe running up to surface of the ground, in your case, you could install a generic threaded plug, or use more expensive surface run off connection point (forget technical name). that most large hardware sale.
with above, just do not connect gutters/downspouts directly to pipe and say good enough. or you may end wasting a bunch of cash and time. due to a clog. it may be next heavy wind/rain storm. or it could be 10 plus years from now. but when that clog happens, you will be SOL! you just have to much length of pipe. not to put something extra in.
on a different note: putting the pipe "deeper" into the ground. will more likely "save" the pipe long term for driving vehicles / tractors, etc... across that area. for myself this is a big one, due to grain trucks to dump trucks with 16ton plus of rock or dirt in them, to other running over my property. i rather install deeper, so there is no issue. and not having to warn someone of being careful in a given area when they will be driving something heavy across the property.
with length of run of pipe, you might be better off getting a ditch witch, "RIDE ON" machine, were you physically set on the machine. and has a chain saw blade but for dirt on one end and a backhoe on the rear end. and a backfill blade on it as well. or just renting a mini EX (small excavator). most small size trenchers just do not dig a wide enough trench for 4" pipe.
