Looking at the second picture, it looks to me like you most definitely have something like LDF4-50A, like what is on this link:
LDF4-50A 1/2" Andrew Heliax Coaxial Cable, Standard
What follows, I didn't expect to be so long, but given hanging antennas and coax on towers was my job for the past few years, I wanted to share as much info as I could.
Preface
That splice is so scary looking my first thought is to stop all transmit operations on that line until it gets fixed. I'd also see if you can find any hams in the area with experience working with coax like that. I used to attach connectors to LDF4 and LDF5 (also called "7/8ths") all the time at my last job, though we had all the prep tools to make it very easy. Andrews is not the only company that makes connectors for LDF4, there's also Eupen and I believe some others, and they all should work with that cable. If you buy them new it is more expensive but they will come with instructions for proper installation which helps a lot if you are doing it for the first time.
I'd be willing to offer assistance myself but I'm afraid that I live a bit too far away over in Florida to do so. My old boss was a ham himself and would occasionally offer assistance installing antennas and coax for clubs if we happened to be doing work on a tower they wanted to install a repeater on. If you happen to have any local amateur repeaters in your area, you might benefit from asking the clubs that operate them who put the antennas and coax up. They might be willing to help you out.
My Recommendations
Coax and Connectors
If money were no object, i'd replace those three sections of coax with one continuous piece of the best coax you can find. LDF4 has an attenuation of about .449db per 100 feet at 150mhz, so with 300 feet you are looking at losses of maybe 1.5db or less, which is practically nothing. You should not be having that much loss that stuff starts melting. When you have multiple pieces put together like that you introduce more points of failure, and each connector will introduce a little bit of loss in your line. It doesn't help matters that a lot of those connectors you have are UHF, which introduces an even greater mismatch than more modern connectors like N-type.
If you choose to keep the coax you have on hand, be sure to inspect every foot of it for signs of kinks or crush damage. I bet the connector at the top of that mast was installed improperly too. Try to make all the connectors N type if you can, with one side of the connection a male and the other female. This way you avoid having to use barrels to join the coax pieces together.
Weatherproofing
As for weatherproofing, the pro's generally use butyl rubber tape as an extra layer of sealant in addition to electrical tape. The way I do it is to first apply a layer of tape over the connectors to keep the butyl from gumming up the connection. We call this "courtesy tape" because it makes life much easier for anyone who has to remove the weatherproofing in the future. Next, I apply a layer of butyl to completely cover up the connection, including the courtesy tape, and smooth it out with my hands as best I can. Then I apply a layer of electrical tape at least 5 or 6 layers thick, as neatly as possible to seal in the butyl so it doesn't bust out in hot weather. The tape holds the butyl in place, and the butyl does an excellent job of keeping the water out. The problem with electrical tape is that over time the adhesive can fail, and water will then seep underneath it and get into your connectors. Butyl doesn't have that issue, but needs tape to hold it in place because it can slowly flow away in the hot sun. One last tip: Never break electrical tape with your hands, always cut it with a knife or scissors. If you break it by hand, you risk stretching it at the end, and that end will slowly pull back and separate from the surface it was stuck to, leaving a tag blowing in the wind.
I've attached a picture showing what the final product looks like, though I don't have any work-in-progress pics available, unfortunately.
Protecting the Coax
Finally, with a run that long I'd be concerned about protecting it from being run over and damaged if it is just lying on the ground like I think it is. I recommend either burying it or suspending it above the ground. Burial might not be an option depending on what type of soil you are dealing with and there is always the risk that someone digging in the future might accidentally cut it. Over time the soil, if it is in a high traffic area, might crush it too, if not buried deep enough.
My preference is suspending it. If you suspend it you need to make sure that you have it hanging from something that bears the load, such as a messenger cable made of steel cable. If corrugated coax is unsupported it can sag under its own weight over the years, stretching out the outer corrugated conductor. The center conductor won't stretch as much, resulting in it possibly being pulled away from where it mates with the center pin in your connector, breaking the connection. If suspended on a messenger cable make sure whatever you use to hold the coax to the cable is wide enough to spread out the weight and not pinch it.
Conclusion
I hope I didn't throw too much info out there at once for you. If you have any questions about what I've written i'll be happy to answer them.
