Welcome to the forum!
I moved your post to Galaxy Gio section, as it is a more suitable place.

Samsung recently has announced that they will update more Galaxy phones to v2.3, including Gio. They however didn't provide a certain timeline (hopefully in summer).

 

When the update is released you usually update in the settings on your phone. Under "about phone", or something similar you will find something about software update.

 

My Samsung GIO had Android firmware 2.3.3. Tried to downgrade to firmware version 2.2.1 with Odin_Multi_Downloader_v4.42 .
During the process my phone is turned off but didn't turned on again.

Now, I have tried many times but it does not want to be turned on in any mode - in normal or in Recovery, or in the Download mode.

Odin finds it but when I try to repeat the process it hangs on the phrase
startup connection

What will advise? How to reanimate it?

 

Unless your phone boots in Download mode, you might not be able to successfully flash a ROM.

The good part is that your phone is still detected as a USB device. Try with ADB to boot it into download mode.

Install and Run ADB and try this command "ADB REBOOT DOWNLOAD"