David - Some answers to your questions:
Do I "need" an electric braking system?
No, but it would certainly help. The brakes on a Speedster (assuming disk/front, drum/rear) are OK for the Speedster, but would be tested a bit hauling a lot of weight around. You want to be coming down out of the Rockies and suddenly have no brakes?
Total Weight, size? (Not more than...?)
That Speedster-looking trailer at Carlisle weighed (empty) about 800 lbs. That's a lot, but his Speedster would pull it (no, it didn't have trailer brakes, but he lives in Ohio, w/o a lot of hills). For a 1915 engine, I wouldn't go much over 800 lbs. TOTAL GROSS WEIGHT, and for that I'd probably spring for electric trailer brakes.
Tom DeWalt (he used to post on here - Anyone have a current email address for Tom??) had been looking into a very small teardrop trailer for his Speedster/Coupe, but then he changed jobs and location and the planning didn't go anywhere. Try contacting him for info - he's someone I would trust on this, as he used to sell trailers. Also, there are a BUNCH of light-weight trailers for motorcycles (Harley and Gold Wing) that might be perfect for you. Some have electric brakes, too.
Questions about my car:
I now have a 1915 engine, is that enough engine, for me to tow this teardrop around the country. (What about going up those huge mountains?)
In my opinion, a 1915 is at the bottom end of displacement for pulling a trailer, but, hell.....people used to trailer stuff with VW sedans at 1600 cc's or less. Just use your head when trailering, that's all...
And those "huge mountains"?? That's what lower gears are for, but remember that, if the mountain road is steep, then you'll end up crawling along at a significantly lower speed (and maybe getting in the way of others).
I've got an extra oil sump, and oil cooler. What worries about driving in over 100 degree days? Or overheating due to climbing hills or going through deserts?
That SHOULD be a big concern. Avoid climbing hills on days over 95F. Make sure your oil cooler has an electric fan on it and that it's working efficiently. ALWAYS keep your rpm's at 3000 or above in order to keep the fan moving enough air to properly cool the heads. Sure, you may make a little noise around town, but so what? Have a calibrated oil temp gauge, as well as a cylinder head temp gauge on cyl #3 (the hottest of the four) and know the limits you should live under.
Other concerns...
Find an old VW Sedan trailer hitch that attached to the frame and have someone adapt it to your Speedster. Talk with others who have done this, or check it out on the Shoptalk Forums. The attachment on a Speedster is different than on a Sedan and should be given some serious thought, but the front should attach to the transmission frame horns.
I am not, personally, a big fan of rear-justified trailer wheels/axles. Positioning them just a bit off-center to the rear should be fine and then you can adjust weight within the trailer to optimum. Put the wheels too far back and most of the weight goes on the tongue and your car making it lower in the back. Put them too far forward and it becomes wallow-y (fish-tails back and forth while moving).
People put trailers on Speedsters. They seem to do just fine with them. All you have to do is research things a bit, talk with people who've done it and learn from their experiences.
Gordon