You don't have to take down the mast to replace your existing halyards. Adding new ones will require taking down the mast or going up in a bosun's chair, an adventure in itself but always a good experience. To replace your old halyards you can use them as a "messengers" to pull the new ones through. Like a fish tape. I like using the tail to pull the new halyard through from front to back. If there's a shackle, just cut if off a foot or two back so you'll have a nice tight piece of rope to connect to the new halyard. To connect them together, and there are a few methods people use to do this, you can sew the ends together....not too tight, because you want the assembly to bend over the sheave as if goes over the top. So 4 or 5 laps of heavy thread, then cover the connection by wrapping tape around it. I've used masking tape, painter's tape, plastic electrical tape... they all work. The point is to keep it flexible and NOT bulky so it will slide over the wheel and down to the exit block at the bottom. When it gets down there be really careful, because those sheaves will be in a metal box, most likely, and will make a fairly tight turn to exit. I will mention that I've also use a paper clip to connect the ends. Again, no problems.
If you decide to add a second jib halyard or a spinnaker halyard you'll either drop the mast or take a ride to the top with someone hoisting you up in a bosun's chair. A good experience, just follow the safety rules. There's plenty of advice on that activity on YT and in this forum. Essentially, you go up, drop in the messenger, tie on your halyard down below and pull it back thrugh. When I did this with a friend, I used fishing line with a sinker attached to keep the line moving downward. At the bottom I used a piece of wire with a hook bent on the end to snag the weight from the mast's interior. After that.... easy.
The "deck organizer in your photo appears to have at least one broken sheave (the plastic wheels inside the housing). You actually need to remedy that, it makes hoisting sails difficult plus it'll damage the line. You may be able to find replacements and they aren't difficult to change out. If you need a full replacement I recommend
Garhauer Marine as a good source for sailboat hardware. good prices.
Finally, I would not try to recreate the existing set up. That right side halyard looks nearly inoperable. The only reason the previous owner had it like that was to use the clutch (or cleat) on the left side. It's the kind of thing that reflects badly on the owner's seamanship. That's why you checked in here....... right

? Well, my friend, if you read a few books, study the marine catalogues and associate yourself with folks who know what they're doing and are happy to share their knowledge then you'll have that boat squared away in no time and become as salty as the rest of us. So......... to fix that issue with the right (starboard) side halyard. Order a simple "horn cleat". A 6 inch cleat is perfect for up to 3/8" line. Mount it vertically on the mast about waist or chest height. pull the halyard out of the left side, and use the new cleat to secure the line and organize the left over, A horn cleat is so easy to work with. Once you have the sail hoisted as high as you can get it by hand, loop it over the cleat and while pulling down again, use your other hand to pull out on the lower part. This little maneuver( called "sweating the halyard" ) will give you extra purchase to raise the sail a bit more.
Now.. if you still have trouble getting the sail fully hoisted without a winch, here's what you do:
1.
release all tension from the boom.... do this anytime, it will make life easier. 2.
lubricate the sail track. step one clean the track A spare sail track slug works best...tie on a light piece of cloth, like tee shirt, squirt it with wd 40 ( I'll get comments on the WD 40) tie on the halyard for pulling up and a length of light cord to pull down.... then up and down you go. Next...squirt some dry, silicone spray lubricant directly on the first few feet of track... then work your sail slug up and down again. Before you hoist the sail spray lube on the track and all the slugs on the sail will pick up some slippery stuff. Shouldn't have any problems now.
I always hoist my mainsail before I leave the dock. Unless there are circumstances, such as being parked in a downwind slip, you should be able to do the same. The point is to get the mainsail up a soon as you can... and do it when and where conditionsare benign, Just remember to allow the boom to swing free so the boat doesn't start sailing before you're ready.
Okay, my friend... I ended up writing a novel here... sorry. If you're looking to start a small sailing library, my favorite recommendation is John Rousmaniere's "The Annapolis Book of Seamanship" He also published a great video series back then that'll really help develop your skills and stoke your enthusiasm. The book's been out for a long time, so older editions will be inexpensive, Check used bookstores for older less expensive editions, or even the local library if you just want to read. Good luck, have fun.