top menu depth level

all you need to do for spotrtal template to make standard drop-down menu instead of default one is to change "Menu Tag ID" parameter in menu module settings from nav to nav2

nav2 is already input.
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The point is, why does it work in local environment but not in internet using absolutely the same configuration.

There is no reason why not, send link so we could have an look

Here is the link:
joomla3.tsv-wolfschlugen.de/handball/

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The menu is defined as follows:
Home (Alias: home)
Beiträge » Haupteinträge
Externe URL
|— Männliche A Jgd
— Externe URL
|—|— A-Jugend 1
—— Externe URL
|—|—|— A-Jugend 2
——— Externe URL

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In the current variant following files have been replaced as you described above.
/templates/hot_sportal/css/template_css.php
/templates/hot_sportal/css/template_css.css

If these files are used in the original variant, the submenu appears with 2nd level but 3rd doesn't appear.

Disable jQuery in hot carousel pro settings

Load jQuery is disabled, now.
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It has no effect on appearance of SubMenu.
SubMenu isn't displayed.