To paint or not to paint the underside

Set it outside during a deep freeze, then put it on the lift and use a needle scaler to remove the old undercoat. Comes off easier this way than melting/scraping. Also tends to flake off more, leaving less residue. You can do the final cleanup with chemicals and a torch.

I wouldn't blast - you'll have sand in the rockers and every blind hole for the rest of history. It can be done well, and done right, but is not something to have done casually IMO.

What you do after depends on what your goals are. Daily driver? Undercoat it, truck bed line, something. Occasional driver that will get wrenched on more than anything? Just paint it and let it ride. It'll get dirty, but will be less of a challenge to clean off after. If a daily, you'll want more protection from rust and chemical attacks. If you plan to never drive it in bad weather, less need for the additional sealing.

Just my $.02 - do what you think fits your goals and desires best. If it were me, I know I'd turn the underside restoration into too big of a project on it's own, and so I think I'd just focus on the engine bay by doing some hand sanding/polishing, and using rust converters here 'n there to clean up ugly spots prior to re-applying paint or spray-on asphalt undercoat. That way it stays a manageable project and will help speed things along faster. But that's just me! I don't build restorations...