Oil and coolant loss...head gasket suspected

So, new here. Searched around and couldn't find exactly what I'm looking here.
Have an 05 gc with 3.7. The heater core has gone out, twice. Never hit the HOT side but it did get hot, both times.
I have oil loss, and coolant loss. Neither on the ground. Oil seems clear of coolant. No coffee coloring or white "bubbles ". Coolant doesn't show sings of oil.
I'm thinking head gaskets are shot. It dies smoke pretty good, with a weird smell, when starting. After say 10 minutes, smoking clears. Sitting at red lights a notice smoke/steam, just briefly.
Also have to replace the conductor plate( seems straight forward).
My sons turning 16 next year so figured we would fix this. Other than head gaskets what should I look for?
We are planning on getting the heads rebuilt . Heard the stock ones had issues.

 

We had the same issue with my sons LJ. Had been running good. Got stuck in Atlanta traffic one day on the way to Athens. It never overheated but it did get a little hot. After that incident coolant started disappearing. Lots of white smoke on startup that would go away after 10 minutes or so. No leaks in the engine bay or on the ground. I bought an upgraded new head from Clearwater Cylinder head in Florida. The head was really nice and fit perfectly. Got a little HP bump with it. No leaks and running good. Coolant is maintaining now. So I assume my head was warped. Had it rebuilt by a local machine shop and sold it. It Could have just been the gasket. But by the way the coolant appeared to be seeping into the engine it seemed like a warped head.
I hope yours is just a gasket and all is well after a gasket job.

 

Our oil did have coolant in it. And we had brown sludge in the radiator.

 

So if you don’t mind possibly having to remove the head twice I would start with replacing the gaskets first. If you are rebuilding the head at the same time I think there is a good chance that will resolve your problem.

 

Thanks. Thinking that if I pull them, they'll get rebuilt . That way I know they are good and the gaskets are fresh.

 

So update...been a slow process. Got the heads out. Piston tops are covered in whatever covers the. Driver side definitely worse then passenger. Heads rebuilt and ready to go back in..... soon, what's good to clean the pistol tops with? Got some flash rust on the mating surface of the block. Asking around, everyone e seems to agree the permatorque gaskets are the way to go for head gaskets... any advice?

 

Ok.....so update...motor is back together. Looks like everything, except intake , is back in its place, plugged in or hooked up. . ... put the battery back in ( probably needs to be charged) turned the key to prime the fuel..didn't really here anything. Dash lit up, radio works AND wipers were on. But they are in the off position. Decided I'd try to start it.... everything went dead. The wipers threw me. Never had that happen. Everything else points to battery. Any suggestions on what else to look for would be appreciated