Catalina installer doesn't open

More space is needed by Catalina because it is about 2GBs larger than El Capitan through Mojave. RAM requirements have not changed. That's still 4GBs. macOS requires an amount of storage space equal to the amount of RAM installed. This is needed to hold the RAM's contents during sleep to save power usage. If you have 8GBs of RAM installed, then that requirement increases from 4GBs to 8GBs.

Installation needs are essentially based on how large the installation disk image is. Catalina's is roughly 8GBs. So, you need 8GBs for the disk image; 8GBs is needed for newly installed system; and 8GBs is needed for the old system before it is deleted. This added space is needed during installation in case there is a problem. Then the installer can remove the new files and put back the old files. When the installation is completed only the newly installed system is left using up storage space. The other 16GBs of storage space is returned to the disk as available space. Therefore, to do an installation the total storage needs will be roughly three times the size of the downloaded disk image. If you don't save a copy of the installer, then it is deleted automatically at the end of a successful installation.

It may be a challenge to free up the needed space but you should not let free space get so low. If you keep adding new files to the disk until it is full, then bad things will happen. The main bad thing is the loss of everything, unless you have current backups. My personal recommendation is that 20-30GBs is the minimum amount of free space you should always maintain. If you haven't enough space now, then you need an external storage drive on which you can store most of your files in order to maintain at least the minimum free space. This may help:

How to Free Up Space on The Hard Drive

1. You can remove data from your Home folder except for the /Home/Library/ folder.
2. Visit The XLab FAQs and read the FAQ on freeing up space on your hard drive.
3. Also, see Freeing space on your Mac OS X startup disk.
4. What is 'Other' and 'Purgeable' in About This Mac?
5. Files That Make Up the 'Other' Storage Category, and How to Remove Them
6. Free up storage space on your Mac.
7. See Where did my Disk Space go?.
8. Be sure to Empty the Trash to recover the space.
9. Replace the drive with a larger one. Check out OWC for drives, tutorials, and toolkits.
10. Use OmniDiskSweeper or GrandPerspective to search your drive for large files and where they are located.

Whatever is there should be put in the Trash. Try downloading again following the instructions in the link I provided. Within that document is the link from downloading the installer. It will appear in the Applications folder as Install macOS Catalina. The size should be 8.24GBs.

No, that is not enough free space. To download and install Catalina will require about 30GBs. Most of that space will not be needed post-installation. However, macOS has a free space requirement of 20-30GBs to allow for OS needs.

Depending upon your Internet speed and connection, it should not take more than 2 hours to download the installer. But you have very little free space which may result in longer download times.

Actually, you still needed the same requirements. Note, the requirements are for installation. After installation, you return to normal. In your case, you were upgrading an existing system, so most of the upgraded files are overwritten by new files with little if any change in space needs. Thus, you are where you should be. Alls well in Stuartville. 👍

The installer downloads into your Applications folder. Try this: How to upgrade to macOS Catalina.