Quarter round and shoe molding are two different things.
Quarter round is just that...one quarter of a circle. Shoe molding, on the other hand, is much flatter with a rounded top.
Lots of people install quarter round when they intended to install shoe molding instead.
Choose shoe molding over quarter round whenever possible, as it looks better.
As for your question, it sounds like your baseboard and shoe molding is 'floating' above the floor. The best solution would be to simply remove the baseboard and molding and put up new stuff (or if the original stuff is of a particularly good finish/quality, then out it back)
If you don't want to remove the baseboard itself, you could remove the quarter round and then look for trim pieces that can act like a shoe molding but a bit taller so that they cover the gap. Visit your local big-box hardware store and you'll find an entire aisle of molding styles to choose from.
If the gap is especially large, you may want to use a molding to cover the gap, then a shoe molding at the bottom of that to create a built-up baseboard.
Definitely the Brad nails. The finish nails will leave larger holes (16 gauge vs 18 gauge usually) and will be more likely to split the wood. Based on the size of your shoe, i dont think you could use pin nails (18-23 gauge).
The Big Box home improvement store always sell kits with finish, brad, and pin nailers for pretty cheap. Some have 2 guns, some 3, and some come with other stuff. I've always found that the best way to go.
Just did a quick check for some kits. Home depot 3 piece Kit for $99. I have had good luck with the Husky Brand from HD. Lowe's had some kits but they were name brand and a lot more expensive. Could also check out Harbor Freight, their stuff is definitely cheaper quality, but maybe that is ok for your situation and usage.
Best Answer
Shoe molding may have to be removed in the future to work on the floor or the wall. That's fairly easy with nails. It's fairly difficult with glue, unless you use something like hide glue that's specifically intended to allow disassembly ... and the techniques used for that would be hard to apply in situ to parts of a house.
Generally, don't glue anything that may need to be serviced, unless you're willing to risk having to destroy it to remove it.