(?i) for case-insensitive pattern would improve many examples.

Faibbus Could i use this for single group? Something like that: (?<name>[\w-[A-Z]]+)\.(?i)(?<extension>jpe?g|png|bmp)

Well, (?i) simplifies the regex in most cases, and yes, maybe the proposed solution isn't the shortest:
The pattern on "Exercise 5 - Search Image Files" can be simplified to Pattern_Exercise5=@"(?i)[\w\.+_\-=\(\)]+\.(jpe?g|gif|png|bmp)";

And "Exercise 6 - Image Files with Path" to:
public static string DrivePattern =@"(?<drive>\b[a-z])";
public static string DirPattern =@"[a-z0-9\-+_=\(\)]+";
public static string DirsPattern =@"(?<path>(?:" + DirPattern + @"\\)*)";
public static string TextPattern = @"[a-z0-9\.\-+_=\(\)]+";
public static string FilePattern = @"(?<name>" + TextPattern + @"\.(?:jpe?g|png|bmp|gif)\b)";
public static string Pattern_Exercise6 = "(?i)"+DrivePattern+@":\\"+DirsPattern+FilePattern;

The thing is that in many languages is easier to declare case-insensitive matches in other ways (Like RegexOptions.IgnoreCase in C#, /i in bash, etc).
For me it's clearer to use a parameter option than the (?i) operator.

Very good exercise, thank you!

Do python regular expressions work the same way

You're saying that metacharacters ^ and $ match start and end of string, but they actually match start and end of lines.

The task description is unclear for some exercises and a correct solution can only be achieved by inspecting the reference matches. E.g., the task description of Exercise 5 states that valid extensions for images are jpg,jpeg,png,bmp,gif (all lowercase). The reference matches indicate that also uppercase letters are allowed.

Because the last description line is important:
"//The filesystem is case insensitive, so name and extension can be on any case."

Both "filename" and "extension" are case insensitive. I can't write jpg,Jpg,jPg,JPg, etc.. with all possible combinations of lower and uppercase, it's not practical. So the exercise just declares the filesystem as case insensitive (like NTFS or FAT32 real filesystems])

exercise 3 works also without character set substraction ...
vowels in the first position are not tested by the validators but the exercise says only consonants in 1st position

Why does the expression @"<[a-zA-Z=\s""-_:]+?>" also match tags starting with a question sign?

yes this is very strange i had to add a [^?] to not match those tags ... no idea why it thinks that it should match it

found it ... the '-' also needs to be escaped ...

The '-' is tricky inside character sets :)
if you use it in the middle of any other two normal characters it means a range, so the expression @"<[a-zA-Z=\s""-_:]+?>" considers all '-' as range metacharacter. Your expression is:
a-z --> A range from 'a' to 'z'
A-Z --> A range from 'A' to 'Z'
= --> The equal character
\s --> Space
"-_ --> A range from '"double quote" to "underscore"
: --> Colon character

You need to either escape it or place it last.

remember a-zA-Z0-9
and /? in the start after <