The basic Italian keyboard layout as shipped with Windows 7 has no way of typing the backtick (`) or the tilde (~). I checked this using Microsoft Keyboard Layout Creator (MSKLC), with that layout loaded into it. I presume that this layout is more or less standard in Italy, though of course Microsoft might have its own oddities here.
However, in Windows 7, there is a somewhat different layout called “Italian (142)”. In it, the backtick can be typed using AltGr + and the tilde using AltGr §. Here “+” and “§” refer to the keys labeled so in the picture in the question, i.e. two keys to the right of “P” and three keys to the right of “L”. I suppose this “Italian (142)” might be some kind of “Italian programmer’s keyboard”, or just a variant keyboard, possibly reflecting different physical keyboards.
If you are using Windows (as I guess because you mentioned “Control Panel”), consider downloading MSKLC and using it to create a modified Italian keyboard layout that suits your needs, and use it as the normal layout, with no need for switching between layouts. You could e.g. make AltGr ' produce the backtick and AltGr ^ produce the tilde; these should be relatively easy to remember due to similarity of characters.
As to the “why” question (why basic Italian layout lacks those characters), I would say that keyboards are primarily designed for typing texts in natural languages, and Italian has little use for those characters. The layout has keys for à, è, ì, ò, ù, so there is no need for a backtick key acting as a dead key (diacritic key) for typing vowel + grave accent combinations, as in many other European keyboards. And while other Romance languages have letters with a tilde, like ã and ñ, Italian does not.
I see that the other answer was already selected, but here is my personal solution. I created a number of hotstrings. e.g. "a will give ä
:?C*:``a::à
:?C*:``i::ì
:?C*:``e::è
:?C*:``o::ò
:?C*:``u::ù
:?C*:``A::À
:?C*:``I::Ì
:?C*:``E::È
:?C*:``O::Ò
:?C*:``U::Ù
:?C*:^ :: ; Turn ^{Space} into neutral ^, else ^ will be used in next vowel.
Send, {^}{Space}{BackSpace}
Return
:?C*:^a::â
:?C*:^i::î
:?C*:^e::ê
:?C*:^o::ô
:?C*:^u::û
:?C*:^A::Â
:?C*:^I::Î
:?C*:^E::Ê
:?C*:^O::Ô
:?C*:^U::Û
:?C*:`" :: ; Turn "{Space} into neutral ", else " will be used in next vowel.
Send, +{'}{Space}{BackSpace}
Return
:?C*:`"a::ä
:?C*:`"i::ï
:?C*:`"e::ë
:?C*:`"o::ö
:?C*:`"u::ü
:?C*:`"A::Ä
:?C*:`"I::Ï
:?C*:`"E::Ë
:?C*:`"O::Ö
:?C*:`"U::Ü
:?C*:' :: ; Turn '{Space} into neutral ', else ' will be used in next vowel.
Send, {'}{Space}{BackSpace}
Return
:?C*:`'a::á
:?C*:`'i::í
:?C*:`'e::é
:?C*:`'o::ó
:?C*:`'u::ú
:?C*:`'A::Á
:?C*:`'I::Í
:?C*:`'E::É
:?C*:`'O::Ó
:?C*:`'U::Ú
:?C*:`'c::ç
:?C*:`'C::Ç
:?C*:ss\::ß
:?C*:ae\::æ
:?C*:AE\::Æ
:?C*:oe\::œ
:?C*:OE\::Œ
Best Answer
There are several keys on a German layout (QWERTZ) keyboard which don't exist on an American layout (QWERTY) keyboard. In order to regain the ability to use the various disparate features, you would need to utilize software capable of remapping keys. If you're in Windows, AutoHotKey does a fair job of easily allowing custom keystrokes.