Is a trike an answer to general doderiness? Yes - if it's a recumbent. (Upright trikes are surprisingly unstable at even low speeds)
Are they easy to ride? Yes - no balance required at low speeds. At higher speeds, and depending on the trike configuration, you may need to learn to lean into corners to stop it lifting a front wheel. (Tadpole)
Tadpole or Delta? Different beasts. Tadpoles tend to be lower and faster so may not be ideal for some people - especially getting on and off. Deltas tend to be higher, so easier to get on and off - but less stable at speed. They also tend to be a bit more complicated if they have a rear drive and differential.
Are trikes as fast as upwrongs? On the flat, about the same. Downhill, I feel a lot safer going downhill at 35mph on my trike than I did going down the same hill at 25mph on my upwrong.
Uphill though they are slower than an upwrong.
At my fittest I could average 15 - 16mph over 30 miles or so on my trike and I am never going to be a fast cyclist.
Traffic? As everyone else has said, motorists notice you and tend to overtake wider and hold back more readily.
Gears? Fit a triple MTB crankset and cassette and off you go. I recommend a shorter crank - say 152mm, which has improved my speed and range considerably.
Cheap options? Depends what you call cheap. You can pick up a well maintained older model for £400 or so. In this bracket you might find an old Trice, a TW Bents or more likely a KMX. Trice were predecessors to the ICE models and the basic design was pretty good and easily evolved into an excellent range - some of the earlier models were rather weak towards the back end and may have seen repairs by now. My first trike was a Trice and £400 well spent. TW Bents were a budget brand and seem quite reasonable. KMX are slightly more of a Go Kart trike. Described as being a bit agricultural, not very refined - but robust and often very good value.
Home built machines are often offered cheap and it is usually quite clear why.
It may be worth a look if there's one near enough and cheap enough - but I would stay clear as too much of a gamble unless you can ascertain the quality of the frame build.
Quality makes such as later ICES will always fetch a good price and you won't see much for less than a grand.
Alternatively you can go the budget new route as I did. A1 Performer Trike E - brand new shipped to your door in a partially assembled state for around £1000 if purchased off
eBay (I was lucky and did not get charged import duty - but reckon for this in your price and you are looking at £1250 ish.
Alternative is to go to someone like Kevin at D-tek in Cambridgeshire and spend the same sort of money for a well vetted 2nd hand model. (I must admit that I have only ever spoken with him on the phone - and suspect that if I ever visit the shop I will leave in a much poorer state!)
All the above options will work better for some than for others - but a starter trike can be purchased cheaply enough and sold just as easily for a similar price.
If you are considering one, it definitely makes sense to take a test ride on a few and see what suits. I would certainly happily recommend a Trike E to anyone looking in that price bracket. Anyone that is near enough to me in West Norfolk is welcome to come and have a test run on mine.