I would appreciate any help with the following project idea:
Emergency sump pump - 12 volt
Pellet stove -120 vac - no igniter, just to drive the blower/impeller - it's a Whitfield Advantage II
Deck lighting - LED lighting, probably up to 30 watts at 12 v

Chances are I wouldn't use all 3 at the same time, but I would like to build/get a battery backup that would allow them to run.  I've seen separate backups for each (Basement Watchdog, Stove Sentry, etc.) but I'm thinking it would make sense to have them share a battery pack.

In addition, I'd like to keep them charged via solar panels...BUT...if the sun wasn't shining and the battery level got too low, it would somehow switch over to A/C charging.

Is there a unit/circuit/controller that would do all of that?

I think the Basement Watchdog sump pump backups use about a 60 Ah battery with a plug-in trickle.  The Stove Sentry is 90 Ah for 8 hours of operation.

I looked at the Basement Watchdog and they recommended NOT using a sealed AGM-type battery, but I'm pretty sure the Stove Sentry would use a deep-cycle of some sort.

Would a 100 Ah battery setup be enough, then? To charge/top-off a 100Ah battery setup would take probably a 50 watt solar panel? More?

Are there charge controllers that switch between solar and A/C ?

If your backup battery ran low for whatever reason, you could charge it from your auto's charging system if you had an extensive power outage and everything was accessible - either via alligator clips on the car's battery terminals or just a cigarette lighter plug plugged into the car.

For maintaining the backup battery fully charged 24-7, you might want to look into a Battery Tender Jr. maintainer. You can leave these connected all the time and they won't over charge the battery and dry it out. And the amount of 120 V current they draw is minuscule.

I would also recommend looking at a battery tender Jr to keep the battery "floated" or charged. They automatically do a bulk charge ever2 weeks or so and then go back in to float, you can leave them connected all the time.

I would recommend you start with 2 golf cart batteries wired in series for 12v. It will be really hard to beat the cost and you will have a good capacity in backup. You can get them at Sam's, Costco or Wal-Mart. Then I would connect that battery "bank" to the sump pump and make sure to fuse it appropriately. Then find out what size inverter your stove needs to run, double it with a minimum of 800w continuous but I would recommend 1500w continuous so you could run a fridge or freezer or any single thing you can plug in to it.

Then run another fused line to out near your garage that you can connect your battery bank in parallel to an idling vehicle and your set to run as long as you can leave it idling.

At that point you could also add a solar panel and charge controller to keep it all topped off similarly to what a battery tender would do, but using the sun instead.

Why not just get a 1 or 2 KW Honda generator? They are very light weight, very quiet, and could be used for other things. In the long run it could also be less expensive.

A great point, for backup scenarios when you are present you are almost always better financially with a Honda or similar inverter type genset as your backup powering / charging method. The cost of solar panels and charge controllers adds up fast. Also if you have a potential need for a small portable genset you wouldn't regret one f these. It is important to note an inverter style genset will cost significantly more, but typically burn between 1/2 to 1/8 (under low load) the fuel for the same load and be significantly quieter.

The upside on a battery / inverter setup is noise and automation. If you power goes out while you are gone for 3 days and your sump pump stops working because of a grid power failure, that's bad, but with a battery backup it will run for some amount of time without human intervention.

There are also water-powered sump pump backup systems that use the domestic water pressure to drive the pump directly (obviously not suitable for systained loss of power on private well systems that use electric pumps.)

While solar is useful as a primary power source for intermittent uses like sump pumps in remote off-grid areas, it rarely makes financial sense to use them that way in buildings that are on-grid. You''ll get a far better return on investment to grid-tie the solar array, and charge the backup system batteries from the grid, since you'll get much higher utilization out of your solar panels that way. You need to keep the batteries fully charged to be adequate for the task, and once they're charged the panels are idle (which should be most of the time, if you want any security out of the system.) Grid tied you get some benefit whenever the sun shines (or even full moonlight).

LED lighting is still less efficient than fluorescent technology, but catching up fast- give it a couple of years. It's still more than 10x as expensive making the NPV negative on a 5 year analysis on power savings. They can last pretty much forever though, and while they lose luminosity rapidly at high temp, the startup is very quick when cold (a real weak point of outdoor fluorescent lighting in cold climates.)

I've been looking at the water driven ones. I have public water. And I don't want to mess around with a sump backup that *might* fail.

The goal here was a very small-scale system, something like a 30 Watt solar panel and a 100 Ah battery...An investment of a couple hundred dollars if possible, not a whole grid-tie system. I'd love to just buy a 12 volt transfer pump and design my own sensor for the water level, but I want to be sure that it absolutely, positively will work....we've had power failures during rain storms and the basement doesn't flood badly, but does puddle up around the interior perimeter drain and onto the cement floor.

I'd want to make sure the batteries were always fully charged, hence the dual solar-AC setup, but I haven't found anything that would automatically switch between AC and solar, depending on whether the sun was shining and the batteries needed charging. The bank would also drive some LED outdoor lighting and be able to power a pellet stove inverter.

Maybe it's better to get a 100 AH battery and a simply float charger and keep that plugged in for the pellet stove and sump backup.
Then get a separate battery/solar panel for outdoor accent lighting. If the accent lighting wasn't working, not as big a deal as if the sump didn't work one day.