Hi Hiker Lauren,
Smart for you to do some research and inquire from locals with experience on this super climb. I'm not sure that upper section of Skyline, (PS to tram), which is in state park boundaries, has re-opened, (I didn't see any info on the SP website mentioning the closure). Hopefully someone on this board has done either Skyline, or tram to summit recently, and can post conditions.
I've not been on either trails this year.
For current conditions above 6k ft. (especially on the west side (Idyllwild) of the mt, check San Jac Johns very helpful site.
https://sanjacjon.com/ .
The mtns got a new dusting the middle of last week, and may have covered some of the trail, although the section from ranger Sta to summit is popular, so it should be fairly easy to follow.
The 7k-8,400ft of Skyline can have ice, and there is a good amount of exposure in areas, so you'd need at least micro-spikes, poles, and hopefully some self arrest skills. That upper section doesn't get a lot of sun, but temps have not been down to freezing at night lately.
Your trail running / wilderness experience will be very helpful even on the lower sections, as most areas in SoCal have become very overgrown from a very wet winter. There are no junctions / connectors to Skyline after the 1st mile, except the "short-cuts", (many are now the main trail), which always connect back. There are maps folks have made showing the route, (not an official trail), and it primarily stays on a ridge north of Tahquitz Cyn.
At times, people make tracks that are not the regular route, and can become rather technical, hopefully the main route is visible, and some locals have "booted it in" correctly.
Temps are looking very good for this week. Be prepared to carry a gallon, (3+ liters), for Skyline, as it is a fairly constant, steep trek, (w some easier sections in the middle), and no shade until 7k.
Cell coverage on Skyline is generally good, until you get into the trees around 6k.
There have been a few shady characters on the lower sections of Skyline, but it has been very rare, (pepper / bear spray may give a bit of security). Going on a weekday, you may not see anyone, going on a weekend may be helpful, as there's a much better chance of having company to help w navigation and other things, unless there 1st-timers too, (problem is, almost no one will be able to hang with you, ha!, although best to just keep in a comfortable, steady pace for the long haul).
Plus, you could always take the tram down after you summit San Jac, (or, even just after Skyline), you'd just need to take a cab / uber back to Museum, bring some $). Going all the way down Skyline is a real challenge, as it's steep, rocky, looks different, (can be a challenge staying on main trail), and it gets hot. I realize you've done some ultras and are very fit, but it's a grind, (poles & a good hat can help).
You'll most likely get others to add more good info here on this board. Happy climbing, scott