The closest Renton indoor gun range is Champion Arms. I personally prefer Wade's indoor range in Bellevue (just east of Seattle).
I've only shot clays in the Southend (Puyallup/Sumner), which might be too far away. But I think the "closest" for Renton will be Renton Fish & Game Club. You'll have to decide how far you are willing to travel.
If you want to shoot HK's, bring your own gun. The nearest HK pistol rental is in the South Tacoma/Puyallup area.
Here's a good website listing all available ranges.
As for applying for a CCW permit, you'll have to apply at a county sheriff office. For Seattle/Renton, that would be King County. Here's the website with some info. They mention it may take up to 60 days.
Enjoy your visit.