What should I do to display this service in site menu?
It is displayed automatically after activation. If this didn't happen, then edit Site Menu manually.

After adding custom fields to the e-mail form there appears "Rebuild form template", and so everything, that has been edited in the template, restores to the initial values. What should I do?
Copy all changes to a temporary file, then rebuild form template, and with the help of the temporary file you can change all you want.

Is it possible to create an e-mail form without the required field for e-mail?
Letters come only to the e-mail specified when creating the form. If you do not need sender's e-mail, make this field hidden, and write in its value any address. This can be done when editing form template.

How can I do the following: at least one of two fields is necessary to be filled in when sending a message? Which field to fill in is up to a sender.
It is not possible. It's better to leave one required field.

I try to add e-mail form to Site Menu, but it "sticks" into page design. How can I fix it?
The code of e-mail form should be put on a page (create a new page if necessary), and then the page address in added to Site Menu.

The field "Reply about successful e-mail sending" contains few characters. How can I increase their number?
You can't. Permitted number is enough.

I'd like a client to receive to his e-mail auto answer, that his request has been received. How can I do this?
You can adjust Autoresponder in mail account. There will be no auto answer from uCoz server.

From time to time e-mail form is not displayed on the page, it is displayed after the page have been refreshed. What is the reason?
Perhaps this is because of your computer or browser settings.

By default there is "--select--" in the drop-down list, but a user can ignore it and the form is sent with incorrect data. How can I "make" the system to check up selection of values?
Set an empty value in the drop-down list, then the system will make a user to select a non-empty value.

At the attempt to use e-mail form the system gives out: "Only registered users can leave messages". How can I fix it?
Guests can also use e-mail forms. You just need to properly set group permissions.

After sending e-mail by means of e-mail form it comes nowhere. How can I fix it?
If a letter has nowhere to come, then it doesn't come. Specify correct e-mail.

Is it possible to do the following: when I select a value in a certain field, some values in another field change?
You should use Javascript.