For some of the tasks you listed, drywall, sheet metal, hinge mortises, I'd probably default to hand tools. For the square hole in the floor, I've seen the oscillator do a much neater job that any rotary tool in a homeowner's arsenal. It is faster and can reach the cutting blade into the project without the motor getting in the way. Besides, it's a neat tool that I just gotta have! Replacement blade costs are the chief drawback that I've seen, though I don't how frequently I'd be faced with that.
JMHO

This is also JMHO. Since I've never owned an oscillator, I'm not qualified to help with your choice. However if you do choose a rotary tool, I'd stay away from Dremel. I used mine for cutting holes in drywall, and the dust killed it. I borrowed my neighbor's for doing some fine cutting of wood, and killed that one. I think the motors overheat too easily.

Dremel tools are typically for finer work/arts & crafts type projects where haevy duty use isn't required. Dremel now makes a multi tool. A multi tool should handle the projects you have listed just fine save the routing of the hinges. That's something that can be tasked by hand with a decent chisel and sharp razor knife with careful work/patience if you dont own a router.

Thanks for your feedback, everyone!

I have a Roto-zip ($150) and a Fein Multi-tool ($400 -but there are cheaper ones out there). This is what I've found from spending the last 7 years doing a complete gut and remodel (rebuild mostly) of a 130 year old farm house.

For what you want.
- cutting a square hole out of hardwood floor for an air duct (although nails that aren't visible through the tongue & groove could make this tricky)

Fien tool - the rotozip struggled cutting through hardwood - lots of burning and went very slow - actually I don't think either it a great tool for this job- a good quality jigsaw would be better.

- cutting sheet metal ducts and gutter downspouts
I have not done a lot of metal cutting with either - but I think the oscelating blade would be better. We've done most of that with a metal blade on the compound miter saw -larger projects get cut with a blade backwards in a skill saw - or by hand - bought a nibbler but did not like the way it works (this was for work on outbuildings).

- cutting notches out of tile for backsplash going around back of sink

I found the rotozip worked wonderful for this (need the right bit) - put your tile on a piece of scrap sheetrock so you can keep the whole tile supported (just cut into the sheetrock). Was worth the full purchase price even if I never use it for anything else.

- small routing jobs (hinge mortises)
I would not use the rotozip - it is harder to control than the Fien tool and you really don't want to slip doing this work. For these things DH uses a good wood chisel. Bigger routing jobs get done with the router.

- cutting holes in drywall
Rotozip made way too much dust - best here is a hand saw of some sort. Utility knife and a steady hand does not make near the mess. For small cuts (like outlets) an exacto knife with a 3" serrated blade was our favorite, but we have not found blades like that again. It was pointed enough you could push the blade through the sheetrock and then start cutting.

- detail sanding
Unless you have holes and need a drum type, the oscelating tool wins hands down. However I've got several nice sanders and they work better than the Fien tool for the job. I also have drum sanders for my drill press so I've not had to use the rotozip to sand anything -

- cutting existing trim and baseboard shorter to install a cabinet
Fien tool hands down. Main reason we bought it. There have been so many times in our whole house remodel that the oscelating tool has turned out to be the ONLY tool suitable for the job.

BTW - I second what others have said - dremel is great for the hobbiest (woodworking, arts & crafts) but I've never considered it for construction type house work. I owned one years ago but gave it away at some point.

Cathy

Are you talking about the new dremel Trio that's being demo'd at Home Depot? Anyway, they're both good for different things (so get both!)
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- cutting a square hole out of hardwood floor for an air duct (although nails that aren't visible through the tongue & groove could make this tricky)

drill, then jigsaw or sawzall (btw, those Ridid Fuego's look tempting) - the multi-blades would not hold up for this.

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- cutting sheet metal ducts and gutter downspouts
sawzall w/ right blade, but the multitool is great for some odd cuts (eg. hole into the gutter, etc)

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- cutting notches out of tile for backsplash going around back of sink - dremel or dremel Trio

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- small routing jobs (hinge mortises)
do it by hand, or trim router (a great tool), or maybe the new dremel trio. I'd go with the trim router.

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- cutting holes in drywall
multi-tool (less dust - catch it with a shop-vac)
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- detail sanding
it depends

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- cutting existing trim and baseboard shorter to install a cabinet
two thumbs up for the multi-tool - worth it for this function alone

Wow, these are super helpful responses, everyone. Thank you so much!

Homebound, I meant the standard dremel (not the dremel multi-tool or the new Trio). I'd be interested to hear any reviews of the new Trio, but I think I'm less likely to buy it than an oscillating multi-tool.

Thanks again!