Took my 4 - 1/2 year old on a morning Coal Creek Falls loop starting from Sky Country Trailhead (Old Man's, Bypass, Fred's Railroad, Quarry, Coal Creek Falls, Cave Hole, Nike Horse). Gaia put us at 3.3 miles altogether.

Falls were lovely. We ran into a work party there - thanks WTA! 

We passed a runner at the Bypass / Fred's Railroad intersection who called out to us a few moments later that she'd just seen a large bear as she approached Clay Pit Road. Thankfully we did not meet it ourselves.

A few downed trees, not difficult to pass over or under. Saw just one unleashed dog and no bags of dog poop - yay!

Felt Cougar Mountain was in need of a "trail runner's condition report" after the recent-ish snowfall.  I ran a big counter-clockwise loop from Red Town, and generally found the snow line to be right around 1100 ft; above that, north- and west-facing aspects tended to be a little icier, while south- and east-facing ones were slushy and/or melted out.  Keep in mind, also, that these are the conditions I found in the early morning on a fairly cold day, so your mileage may vary.

For a more detailed, trail-by-trail view, starting from Red Town TH:

Wildside Trail, Marshall's Hill - patchy snow, a few small trees down, generally good running.
De Leo Wall - all melted out, truly excellent running
Far Country, Shy Bear - patchy snow up to the Deceiver junction, but no major issues
Deceiver, Long View, Wilderness Peak Trail - packed snow and ice; use caution on downhill sections.  I'd place a good bet these get sloppy in the afternoon, but do not know for sure.
Fred's Railroad, Klondike Swamp - packed snow, but still pretty good running because the terrain is so flat.  A few trees down, but pretty easy to navigate around
Lost Beagle, Harvey Manning - packed snow and ice; again, use caution on downhill sections.  Harvey Manning was already starting to get sloppy by about 10am on some sun-facing aspects.
Clay Pit Road, Cave Hole Trail - well-packed snow above 1100 feet, these are clearly heavily traveled.  Made for fast running except for a few steeper downhill sections, which were quite icy.  The last mile of Cave Hole back to Red Town was fully melted out.

Happy Running!

Highlight 1: 8 foot bridges over small creeks within a 1 mile hike of Sky Country lot.  Hydrology willing, if you hike (Sky Country Lot -> Sky Country Trail -> left on Military Road Trail -> left on Red Town Creek Trail) by the time you reach Cave Hole Trail you've hiked 1 mile and crossed 8 foot bridges over small creeks (see pic below of fanciest one).

Highlight 2: Coal Creek Falls is always a dazzler (see pic).

Cougar Mountain Trails Map

Hike GPS data for Google Earth viewing

Per Garmin, 5.02 miles/818 vertical feet.

Parking: At about 3:30 PM on a sunny Saturday, Sky Country was mostly full.  Port-a-potty on lot.

Trail: The top of Military Road Trail and some of Coal Creek Falls trail are muddy, but you can just go around it.  There are still some trees down (e.g., top of Quarry) you can't step over, but you can climb over or crawl under them.

Wildlife: Many, many birds including a very boisterous flock of waterfowl on Klondike Marsh.

Highlight: Doughty Falls is beautiful, with a very different character than the already-popular Coal Creek Falls.  I think Doughty is worth adding clambering over some trees and a couple extra miles to your hike.

Cougar Mountain Tails Map

Hike GPS data for Google Earth viewing

Per Garmin, 4.94 miles/663 vertical feet.

Parking: About a dozen cars at Sky Country at 10 AM on a party-cloudy Thursday.  Port-a-potty on lot.

Trails: Frequently muddy but manageable; nothing impassable.

Obstacles: Trees are down all along the south extent of this hike; from Shy Bear Pass in the east to the end of Deceiver in the West.  But, if you can climb over things and crawl under things, you'll be fine.

Wildlife: Things are REALLY waking up.  We heard TWO species of frogs croaking, many birds singing, and we tripped rounds of alert calls from many Douglas squirrels.

Coal Creek Falls: Flowing impressively; trail there from Sky Country a little muddy; no tree obstacles; see picture.

Cougar Mountain Trails Map

Hike GPS data for Google Earth viewing

Per Garmin, 4.03 miles/469 vertical feet.

Parking: Mostly full Sky Country lot at about 11 AM this MLK Monday.  Port-a-potty on lot.

Trail: Muddy in spots, but manageable; otherwise good traction.

Tree Obstacles: One tree on Coyote Creek and another on Quarry (above Coal Creek Falls intersection) are over-or-under tree falls.  The one on Quarry was enough for the grizzled old guy hiking in front of us to want to watch us go over it first.

Wildlife: Some really pretty bird songs.

Caution: Blind corners and drivers disregarding traffic directionality create Sky Country parking lot hazards.  Every stretch of the figure-8-shaped lot is a one-way (see the attached picture).  Directionality is indicated with traffic signs and pavement arrows.  However, people frequently drive against the one-ways.  We’ve all disregarded parking lot traffic signs, but the lot has blind corners that are only safe when directionality is followed.  I have seen people charging excitedly into the parking lot and around blind corners with the confidence they have the right-of-way to the whole lane.  I have also seen people in a hurry to get home drive furiously around the same blind corners the wrong way.  At some point, people will collide.  So, do not be one of those people; recognize that, once you are in the parking lot, Mad Max rules apply and drive cautiously around every corner assuming someone could be hurtling against the one-way toward you.