Now that you have some fire-formed brass you just gotta reload for it. The are several much more comfortable handloads with cast bullets that will make it from pistol to rifle performance. FWIW, this is an open sight rifle 'just because', if you want to scope one get one of the newer versions.
I love Trail Boss and Lee 340 to 405HB cast bullets in my 45-70s, but Unique will do well too. You likely want at least ,459 dia. lead bullets, dont even bother with ,457s. Shikari owners can be more specific on the cast bullet size(s) that work best.
Its a cool old rifle, not a 'collector' but to an H&R NUT (none here
) and it sounds like you respect and care for your stuff already so you wont be hurting it to use it.
On the trigger, just clean the guts up and oil everything but put a dab of automotive anti-sieze compound only on the sear tip and hammer notch, it will likely work nicely with no metal removal. Those old non-transfer bar ones are pretty nice, but often can benefit from a new Wolff mainspring (dbl. leg).