I take my Scout troop to Henry Coe frequently.
Easy at Henry Coe is hard to find once you get out of the Headquarters, Manzanita Point, Frog Lake triangle.
For Coyote Creek to Kelly Lake, my usual route is:
Coyote Creek->Grizzly Gulch Trail->Dexter Trail->Wasno Road->Kelly Lake Trail
Though this route is a little bit flatter and a little longer.
Coyote Creek->Grizzly Gulch Trail->Rock Tower->Jackson Road->Wasno Road->Kelly Lake Trail
The Dexter or Tower trails (depending on the route you choose) are the steepest part of that hike. Both are memorably steep.
There's an iron ranger at Hunting Hollow where you pay.
You can park along the road between the Hunting Hollow entrance and the Coyote Creek Entrance … I know, the signs say no parking but that's misleading. You can park as long as all four wheels of your car are off of the pavement. You'll see people parking like that and I even asked a ranger about it.
The other thing to remember about parking is that they really want enough space to get a fire trucks and heavy equipment into the Coyote Creek entrance, so don't park anywhere where you might be construed as blocking the entrance. People do park at the gate but I try to park along the road when I'm using the Coyote Creek entrance.
Water management is an important skill at Coe but on that hike, it's not too hard – plenty of water at Kelly Lake but not much between the entrance and Kelly Lake.
The trails and roads are virtually all well-marked and obvious.
The roads at Henry Coe double as fire breaks, so they tend to ignore the topography go straight along the ridge lines.
I don't think I've ever seen a mosquito at Henry Coe, irritating little flys yes, but mosquitoes have never been a problem for me. I use a hammock, so I don't have bug netting to protect me and don't worry about it.
Sometimes it gets a little crowded at Coe during the spring. Flat spots are rare at Henry Coe, especially as you go north and west. At Kelly Lake there are several flat spots that you can set up on but you could also consider camping near one of the numerous ponds.
April is a little early for sticker season, but some short gaiters, like REI desert gaiters can make life more comfortable in May.