How do you insert a patch with Hoppe's #9?

I put the barrel into a container that won't damage the crown and fill he container with solvent. I then put a patch on the plastic rod tip and run the rod down the barrel and on the back stroke it sucks the solvent straight up into the barrel. This assures that I get a good cleaning and rinsing of the entire barrel all the way to just where the shoulder begins. I then clean the lug recesses and other areas that can get grime build up. Then I start putting clean patches down the barrel until they come out as clean as when they went in. I give the bolt a rinse in acetone to remove any grit that may inadvertantly get inside, I live and shoot in a the desert. I put a thin coat of oil inside and outside the barrel and put them up until next shooting session. I take a patch saturated with acetone and run it through the barrel before shooting to remove all oil residue.
Years ago I would just run a patch through the barrel with solvent on it and and then a few clean dry ones until they came out clean. But I learned that I was leaving a bunch of copper residue in the barrel and thus started drenching the barrel with solvent and letting it sit for a good 15 minutes.

I use a rod guide "aka bore guide" with a solvent port. No mess, no fuss.

Hoppe's #9 isn't going to bother the action of any rifle.
"...how often to rub a little lube onto the bolt..." When it needs it. And not a whole lot.

Another user of bore-guides here.

I use the Possum Hollow nylon type, and jags with the "pointy tip". Put a dry patch on the jag (piercing the patch with the tip), apply cleaning solvent or oil, then insert the wet patch into the bore guide.

I put a small box under the muzzle that collects the dirty patches when they fall off on the return stroke.

I have a couple of plastic boxes that rubber band onto the barrel at the muzzle end to perform this same job if you're holding the rifle "free hand", but I quit using these when I started using a "gun vise" to hold the rifle on a bench.

The Possum Hollow type has O-rings that do a pretty good job of sealing the bore guide to the chamber. As mentioned already, a powder solvent like Hoppe's No. 9 isn't going to hurt the action. However, it might flush out lubricant that you want to stay in place. Bore guides help the stuff stay where it belongs.

Bore guide is the way to go for me.

They actually come with a number of different "head" pieces for use on different rifles/bores.

I can even remove the headpiece altogether and lock it in place for use on my bolt action .22lrs....very handy.

Gamestalkers method is also very helpful if you lack the equipment we are speaking of..

I use an MTM bore guid and jags. I cut Hoppes with Kroil and store in a glass bottle with a medicine dropper. I use the dropper to wet the patch once it's on the jag and push through the bore guide. Usually it stays on

1) take patch and place over top of bottle....hold in place with thumb or finger
2) shake bottle
3) place patch on tip of jag
4) run through barrel
5) remove patch from jag and remove rod from barrel
6) repeat as desired

dont worry about chemicals in your action......if they are safe to put down the barrel.....chances are pretty darn good its safe to put everywhere else.

I got tierd of spillage (my fault) and now use the spray #9 aresol.

I like to use a sling-shot.

Patch on tipton speer-point rod; dip in bottle of Hoppe's, hope it doesn't fall off, let it drip on floor and table as I move it toward the breech, push through bore, let it fall off the other end onto the kitchen table or floor on the return, listen to wife yell about the mess & complain about the smell, I smile a little wider, soak another patch, run another through, more yelling, but blocking it out now, next patch......