How many characters can be mapped with Unicode?

Unicode allows for 17 planes, each of 65,536 possible characters (or 'code points'). This gives a total of 1,114,112 possible characters. At present, only about 10% of this space has been allocated.

The precise details of how these code points are encoded differ with the encoding, but your question makes it sound like you are thinking of UTF-8. The reason for restrictions on the continuation bytes are presumably so it is easy to find the beginning of the next character (as continuation characters are always of the form 10xxxxxx, but the starting byte can never be of this form).

Unicode supports 1,114,112 code points. There are 2048 surrogate code point, giving 1,112,064 scalar values. Of these, there are 66 non-characters, leading to 1,111,998 possible encoded characters (unless I made a calculation error).

I am asking for the count of all the possible valid combinations in Unicode with explanation.

1,111,998: 17 planes × 65,536 characters per plane - 2048 surrogates - 66 noncharacters

Note that UTF-8 and UTF-32 could theoretically encode much more than 17 planes, but the range is restricted based on the limitations of the UTF-16 encoding.

137,929 code points are actually assigned in Unicode 12.1.

I also don't understand why continuation bytes have restrictions even though starting byte of that char clears how long it should be.

The purpose of this restriction in UTF-8 is to make the encoding self-synchronizing.

For a counterexample, consider the Chinese GB 18030 encoding. There, the letter ß is represented as the byte sequence 81 30 89 38, which contains the encoding of the digits 0 and 8. So if you have a string-searching function not designed for this encoding-specific quirk, then a search for the digit 8 will find a false positive within the letter ß.

In UTF-8, this cannot happen, because the non-overlap between lead bytes and trail bytes guarantees that the encoding of a shorter character can never occur within the encoding of a longer character.