Questions about Intel Core i5-12400F

I think there's an error in the description and the details: if it's a i5-12400F is an i5 (not i3 as shown on the picture) and 6-core processor (not 4-core as shown in the description and the specifications).

Is only the CPU included here, or also a CPU cooler?

I wanted to ask if a PC with the following components is possible, and if not which component I should replace: -Kolink Observatory Lite -Intel Core i5-12400F LGA 1700, 2.50 GHz, 6 -Core -B660 DS3H AX Gigabyte B660 DS3H AX LGA 1700, Intel B660, ATX -ASUS DUAL Radeon RX 6650 XT OC Edition 8 GB -Corsair Vengeance RGB Pro 4 x 8GB, 3200 MHz, DDR4 RAM, DIMM -ASUS TUF Gaming LC 120 ARGB -WD Green SN350 1000 GB, M.2 2280 -Thermaltake Smart RGB 600 W Thanks a lot

intel core i5 processor was delivered only in a plastic bag without a box ?????normal???

What does this cache stand for? There is another i5 12400f for 220 that doesn't have a cache. What is the difference?