Reddit mentions of kenable 5 Pin Male Din Plug to 4 x RCA Phono Male Plugs Audio Cable 1.2m (~4 feet)

It seems that your record player is actually a receiver amplifier/tuner combo) with a built-in turntable (i assume it's this model: http://www.hifiengine.com/gallery/images/luxor-dirigent-2x17g.shtml)

It should have an input/output for a tape deck on the back, probably a 5-pin DIN connector (usual for European equipment from the 70s). You should be able to hook it up to your modern audio system using a cable like this: https://www.amazon.com/kenable-Phono-Plugs-Audio-Cable/dp/B008N29OSG

When you're using the turntable, the Luxor will send audio through the tape output. This was used for recording records to tape back in the days, but works just as well for connecting it to other equipment.

Do your VICs have a 2-pin power connector (next to the power switch) or a round power connector with a bunch of pins (up to 7)? If the latter, they use the same style of power supply as the C64. These power supplies eventually fail and when they do, they fry the chips in the computer. You can either get a "saver" device that goes between the power supply and the machine, or you can get a power supply that's safe. (If it's the 2-pin kind you have, they're safe.)

For software, you can get a SD2IEC, and SD card reader that (roughly) emulates a Commodore floppy drive. It'll work on other Commodore 8-bits as well, if you eventually want to try other machines.

For the VIC-20, though, if you just want to get a bunch of games then a great/easy option is to get the Penultimate+ cartridge. It includes more than 70 games, and also has some other useful stuff like a RAM expander. (I have no connection to TFW8B, other than being a satisfied customer.)

If you don't have a joystick, any Commodore or Atari joystick or paddles should work. They are easy to find on eBay. Don't use a Sega Genesis/Megadrive controller -- they work until they fry chips.

The cable with a box and red jack sounds like it may be the RF modulator box. This converts the audio and composite video from from video port into a signal analog TV tuners can use (usually on channel 3). If that's what it is, you won't need it with the monitor. For the monitor you'll probably need a cable that has a DIN connector on one end, and 2 or more RCS connectors on the other end. If you don't have one, you can get one on Amazon. With that cable the black (not red or white) is audio and the yellow is composite video when plugged into a VIC-20.