How low is the pitch........

If it rains but there is no wind does it leak still ? If yes, then wind is not part of equation, otherwise, it's the wind that is pushing the water under the shingles. So, i'm thinking this pitch is very slight or the wind is very strong. How many layers of shingles on roof ? Shingle nails are just so long. May be loose shingles bc nails lost their holding ability thru so many layers

I won't be able to measure it until the weekend, and it's only when it's windy. So I think the safe bet is that its coming up and under.

It looks like a small area, so cost to redo not too much. As a DIY'er What would i do ? Take off all shingles down to the wood. Apply new tar paper, then reshingle it. Has this been happening for a while ? If takened down to the wood then you will see your answer. For wood to rot = (water + oxygen + time)

One layer of shingles except where the pitch changes. (not easy to see in the picture) This room was added on after the fact, and thats where the lower pitch came into play. The roofer came back gratis and doubled up there and used extra nails. The total affected area is ~8x16.

It looks like you have maybe a 3:12, maybe a little under. In my personal and professional opinion it's absurd to put shingles on a 3:12.............especially when it's an add on like yours. I'm betting there is very little air circulation in this add on if any. This will bake the shingles from the underside which shortens the service life. You can tell by looking at the shingles. If they're cupped they're shot., the adhesive stripe on the service edge of the shingle releases..........and some wind to a thunder storm or rain and you've got the perfect recipe for leaks. You might try a roof coating for a temporary fix (roll on rubberized bit).

When you do reroof your home, if the add on is going to stay, I would recommend two things. 1. Add the ventilation to your soffits and use insulation pockets at the wall line to keep the insulation from blocking the air flow. Think about convective heat flow.........hot air rises, so you want soffit vents and outlet vents at the high end of the roof. 2.) Go with a rubberized bitumen granulated roll roofing on this low pitch over over a continuous layer of ice and water shield. Both the underlayment and the finish roll should be self adhering and they'll nail the roll at the over lap edges......................unless you have knuckle heads install it, I can assure you your leak problems will be history.

The temporary part is the whole house. Within 2 years, everything will be completely rebuilt. So while the right long-term thing would be to rip it all off and re-do it with a heavy-duty liner or simply as pure rubber roof, its more than I'd rather do. I'm guessing that the shingles are Flapping Up and the wind-driven rain is going up n under. Would I gain anything (or make it significantly worse) if I used some sort of rubber glue in a chaulk gun and do a dot or 2 under each flap? Not a long line which would keep whatever water in; but just a dot?

Be careful about the advice you are getting. You need to have a roofer look at it.
In our region in the NorthWest, anything under a 4:12 should not be shingled because rain will be pushed up the shingles as you are experiencing.
Did you have a professional reroof it? If not then you need one to come look at it. If not, then that would explain why you are having the issue. Some things are easily enough a DIY task, others (such as the roof) are very critical to the structure and a professional's years of experience are highly valuable. If it was done by a professional and that is less than a 4:12 you have a bad roofer on your hands.
I believe most manufacturers of asphalt shingles specify not to use on less than a 4:12...
You have a lot of temporary options, but I can't suggest anything there as they are all less than professional. But things like Sophie mentioned will work.

I'm not sure if it would solve the problem, but you could try a roof coating. It might stop the water from leaking in long enough to hold you over till you redo it all. A roof coating would also be less expensive than a replacement. http://www.stetsonpainting.com/what-we-do/roof-coating/