Can I use my old Imac 2013 as an external monitor for my macbook?

No. Your 21.5" 2013 iMac would support Target Display Mode, using Thunderbolt input, when running High Sierra or earlier. But your 2020 MacBook is too new to use ANY iMac as a Target Display.

If that restriction didn't exist, you'd still need $80 in specialized Thunderbolt connection hardware to reuse an old 21.5" 1920x1080 display. (Don't spend the $80. You'll probably find that you can make a data connection but not the Target Display one you were hoping for.)

Save the $80 and put it towards a real monitor.

Short answer: no.

Although your 2013 iMac supports Target Display Mode, the other Mac that you're connecting it to must have been introduced in 2019 or earlier and have macOS Catalina or earlier installed. The cable connecting the two Mac computers must be a Thunderbolt or Thunderbolt 2 cable.

Ref: Use your iMac as a display with Target Display Mode - Apple Support

Yes.

I have a 27" iMac 13,2 (late 2012) and a MBP M2 (2023).

Enable screen mirroring on your MB.

• Go to Control Center and click on Screen Mirroring
• Select your iMac and choose whether you want to mirror or extend your desktop.

Alternatively, you can also enable Screen Sharing on your MB and access your MB from your iMac, as long as they are on the same network. You can do this by going to System Settings > Generals > Sharing > Content & Media > Screen Sharing [enable].

Where there's a will, there's a way.

saelmira wrote:

i also wonder if I can download chrome and zoom safely in my old imac.

The Google site indicates that Chrome requires macOS 10.15 (Catalina) or later.

https://www.google.com/chrome/dr/download/

I believe that the current version Firefox will also run on Catalina.

https://www.mozilla.org/en-US/firefox/new/

There's another browser called Brave. I believe it runs on Catalina also.

saelmira wrote:

I just worry about hackers since I have saved passwords to google and the old IMac does not update anymore

Computers can have security vulnerabilities, but many criminals these days will try to hack you with "phishing" messages – because they think it's easier than hacking your machine.

Many of the scam messages will try to prey on your fears of security weaknesses. They'll claim that you have a "Trojan virus", or that your machine is infested with "99 viruses!!!!", or that your "iCloud is hacked!!!!!" All lies – mean to scare you into contacting "tech support" (actually criminals) for "help" (as in help stealing your money).

Occasionally these lies contain a nugget of truth, so if you see a claim that a particular password is compromised, change that password immediately.