To be a little more clear on this point, if the wire itself is too long to support its own weight, increasing the size of the wire will not help, but changing the type of wire may. If you increase the size of the wire to double its strength, you also double its weight. But an antenna wire often needs to support more than its own weight. It can be called upon to support the weight of ice, birds, wind load, baluns, transmission lines, spreaders, antenna insulators, rope (to extend the length to a convenient support point), counterweights (to account for deflection of trees or other supports) or bungee cords, etc. Bigger size probably doesn't really help with ice since a wire that is bigger will probably collect more ice.
The size of the wire does effect the performance of the antenna, even a receive only antenna. Bigger wire may be more broadband, though to get much benefit you may need to increase the apparent wire size by running multiple wires. At higher frequencies skin effect limits the conductivity of the wire. It has been observed that electrically short antennas might be more affected by wire size.
Small wires may be a hazard to birds who can't see them.
The National Electric Code mandates certain minimum sizes for outdoor antennas depending on the wire length and type, ranging from 14 gauge to 10 gauge. This is often ignored as the NEC is messing around in areas way outside their competence.
Now back to the OP's question. Be warned that antennas are an extremely complicated subject. I will not go into much depth.
Using the thicker wire (smaller wire guage number = thicker wire) won't hurt and might help. Length: use the whole roll, leave it on the roll and unwind what you need. Longer is generally better but there are some notable exceptions as antennas have resonances. If you are working indoor use the longest wire that is practical and keep it as high as practical.
A proper antenna for the lowest frequency shortwave band would be two wires slightly shorter than 100ft in opposite directions (a 1/2 wave dipole) and at the highest frequency shortwave (excluding the ham bands above) would be about 9ft in either direction. The most used shortwave band wants an antenna that is a little under 25ft in either direction. A long wire should be 4 times as long (twice the total length).
Unfortunately, a good antenna at one frequency can perform very poorly at other frequencies. It is particularly bad when the wire is close to twice as long as it should be (or at half the frequency it is best at) or any odd multiple of twice as long. A wire that is too short will not only not pick up much signal but will have more of these notches over the bands of interest. Sometimes changing the length of wire may be necessary to pull in a particular station.
And a long wire or random wire antenna, as opposed to a proper doublet, needs a ground connection to work against. Actually a better ground than any you are likely to have available but any ground will probably help.
Shortwave stations often use ridiculous amounts of power (up to 500,000W) to cover a wide area and so people with crappy antennas can pick them up. But some are much lower power.
Consider what you are trying to receive. The big name shortwave broadcast stations? Lower power local regional shortwave stations? Lower power distant regional shortwave stations? Moderate power ham radio stations? Low power (QRP) ham stations? Marine or Aviation HF bands? For the non-shortwave broadcast applications or the lower power more distanct stations you need a much better antenna.
If you get too much signal, add an attenuator. You can throw away excessive signal from a good antenna but you can't really get back signal you have lost due to a crappy one (amplifiers are limited by noise).
Instead of clipping on to your existing antenna, use the antenna connector on the radio if available.
As a quick and dirty shortwave antenna, I have a roll of about a 100ft of hookup wire with an antenna connector on one end that fits the shortwave reciever. I unroll what I need/can fit. The extra wire on the roll provides some end loading (but not optimized) which may reduce the effects of having too short a wire somewhat. If you don't buy one of the premade wind up antennas, a cheap coleman camping clothesline reel or coughlans laundry reel or perhaps even a chalk line reel can be another way to get a wind up reel. If you want to get more serious, use two and make an adjustable dipole.
Here is an example of an engineered random wire shortwave reveive only antenna that gets good reviews. 45 feet long with a matching balun.
http://www.universal-radio.com/catalog/sw_ant/2205.html
Don't forget that lightning can be an issue for antennas, especially outdoor ones.
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