Can I have the same, but when clicking the token B to show the Handout
(no longer showing the link in the chat and having to do this
additional click)?
No, this is currently impossible. To open a handout programatically requires API access. However, you can put a lot of information in that bar, including complete roll templates and images. Most markdown formatting is accepted. Asterisks for bold and italic, or [image name](Image URL)
for image display.
Can I have token A to activate a macro onClick, so that the players do
not have select the token and then to click the macrobar action (I am
not a paid customer at roll20)?
If you use the above method, to display the info in chat, and do not want them to select the token, then use the following code as a universal token action (create the macro in the Collections Tab, click "Show as token action" and make sure macro is usable by "All Players"):
@{target|bar1}
This can also be saved as a universal macro bar action, and it will not require any token to be selected, but you will need to tell the players to click "Show in Macro Bar" in their own Collections tab. Each players Macro Bar is their own setting.
Example:
create a map pin token for the "Temple of Alante". Put this text into Bar 1:
`&{template:default} {{name=@{selected|token_name}}} {{=[x](https://i.imgur.com/TG4w1Ad.jpg) This is the name of the Temple consecrated to the Western gods. It was named to honor the land lost beneath the waves during the Whelming of the West. The current high priestess is named Thesandra, and was appointed by Queen Gloriana of Tar Olan to represent the Western Gods. Thesandra has been charged with advocating a Western Gods seat upon the High Council, but this is a nigh impossible task, since the signatories have yet to admit one more High Council seat, and other races would almost certainly reject a third human representative. }}
This will result in the following output in chat:

There are ways to format the output more beautifully, but this is an out-of-the box solution that will work for any sheet, and at any subscription level.
Note, that if you still want to use handouts, you can put a very short abstract of the subject in Bar one and still include a "for more info, click here" link to the handout. Most players can be overwhelmed by too much info at once, but can click the link if they really want to know more.
Final note: Use a bar that you would not mind other players having access to. For instance, you might use bar1 for monster hit points and not want players to read that info. Since the action above will work on any token, choose carefully.
Much more info and different techniques and approaches can be found here on the Roll20 forums.