Vyxal 3, 21 bytes

Hello World, 2 bytes

kH

Vyxal It Online!

Factorial, 1 byte

!

Vyxal It Online!

Primes, 3 bytes

ɾ~℗

ɾ~℗­⁡​‎‎⁡⁠⁡‏‏​⁡⁠⁡‌⁢​‎‎⁡⁠⁢‏‏​⁡⁠⁡‌⁣​‎‎⁡⁠⁣‏‏​⁡⁠⁡‌­
ɾ    # ‎⁡range 1..implicit input
 ~   # ‎⁢filter by
  ℗  # ‎⁣is prime?
💎

Vyxal It Online!

Unique chars, 3 bytes

ʀu⅟

ʀu⅟­⁡​‎‎⁡⁠⁡‏‏​⁡⁠⁡‌⁢​‎‎⁡⁠⁢‏‏​⁡⁠⁡‌⁣​‎‎⁡⁠⁣‏‏​⁡⁠⁡‌­
ʀ    # ‎⁡to lowercase
 u   # ‎⁢uniques
  ⅟  # ‎⁣strip whitespace
💎

Vyxal It Online!

diamond, 12 bytes

-5 bytes by pacman256

4ƛ★×4«⨪]æᐕJ”

4ƛ★×4«⨪]æᐕJ”­⁡​‎‎⁡⁠⁡‏⁠‎⁡⁠⁢‏⁠‎⁡⁠⁢⁤‏‏​⁡⁠⁡‌⁢​‎‎⁡⁠⁣‏⁠‎⁡⁠⁤‏‏​⁡⁠⁡‌⁣​‎‎⁡⁠⁢⁡‏⁠‎⁡⁠⁢⁢‏⁠‏​⁡⁠⁡‌⁤​‎‎⁡⁠⁢⁣‏⁠‏​⁡⁠⁡‌⁢⁡​‎‎⁡⁠⁣⁡‏⁠‎⁡⁠⁣⁢‏⁠‎⁡⁠⁣⁣‏‏​⁡⁠⁡‌⁢⁢​‎‎⁡⁠⁣⁤‏‏​⁡⁠⁡‌­
4ƛ     ]      # ‎⁡map over 1..4
  ★×          # ‎⁢"*" times n
    4«        # ‎⁣left-pad spaces to length 4
      ⨪       # ‎⁤palindromize
        æᐕJ   # ‎⁢⁡duplicate reverse join
           ”  # ‎⁢⁢join on newlines
💎

Vyxal It Online!

Created with the help of Luminespire.

Ruby -rprime, 390 bytes

I don't know why I decided to try this.

-rprime flag loads the Prime library for task 3. As per the meta post on program flags, adding flags treats it as a "different language", so I can't add -n or -p for some tasks and remove it for others, even if it would save bytes. (Tasks 1 and 5 take no input, and no input means programs with those flags won't run, which is why they can't be used.)

Task 1, 47 bytes

Backslash-escaped almost every letter to free letters up for usage in later tasks. $> is an alias for STDOUT, which will be used later.

$><<"H\x65\154\154\x6f\54\40W\x6f\x72\154\x64!"

Attempt This Online!

Task 2, 31 bytes

3+9+~9 == 2 to avoid reusing 2.

p((3+9+~9..gets.to_i).inject:*)

Attempt This Online!

Task 3, 28 bytes + 56 (().ceis) = 84

Since this is the only task that doesn't say to take input from STDIN, this one is a lambda instead so I don't need to take a huge penalty from doing things with gets.to_i or something similar. This is probably the only time where lambda saves bytes over its stabby syntax ->, which has to buy back > and uses up the - needed in Task 4.

lambda{|e|Prime.each(e).sum}

Attempt This Online!

Task 4, 67 bytes + 96 (*.<achinrstu) 71 bytes + 72 (*.<achrst) = 163 143

This one had wayyyyyy too much overlap but it was needed to do the given task. Uses t,=*STDIN to get the line of input, assuming newlines won't be present. Let me know if that's an incorrect assumption.

t,=*STDIN;t=t.tr'A-Z','a-z';t=t.tr'^a-z','';t=t.chars;t&=t;STDOUT<<t*''

Attempt This Online!

Task 5, 53 bytes + 32 ($*<>) = 85

Buys back virtually every character in the program, since it's still cheaper than any alternative I could think of. Uses heredoc syntax for the string since we no longer have ' or " and we're buying back < anyways since it's cheaper than buying back puts.

$><<<<Q
   *
  ***
 *****
*******
 *****
  ***
   *
Q

Attempt This Online!

Itr, 83 bytes, no characters repeated between programs

The five programs separated by newlines:

"Olssv'^vysk("7-

P

µf«äµL1=×S

32|ÍÌM64á<F¥

»   *
  ***
 *****
*******
 *****
  ***
   *

online interpreter: 1 2 3 4 5

The letters in problem 4 are printed in alphabetical order instead of the order they appear in, I did not find any rule against that

Explanation

"Olssv'^vysk("7- ; Hello World! shifted by 7 to avoid space

P ; product over the 1-based range to the input

µf«äµL1=×S
µf«         ; replace the numbers in the range 1 : input with their prime-factorisation
   äµL1=×   ; only keep the elements with a length of one
         S  ; sum up the array

; assumes that the input consists of ASCII letters and white-spaces 
32|ÍÌM64á<F¥
32|          ; or with 32 -> converts uppercase ascii to lowercase
   ÍÌ        ; get a list of all unique elements
     M64á<F¥ ; print the elements that are larger that 64 (filters out white spaces)
     M       ; for all elements
      64á<   ; is the the number greater than 64
          F  ; repeat that many times
           ¥ ; print

; string-literal containing the required output
; all possibilities to loop are used up by the previous answers
»   *      
  ***
 *****
*******
 *****
  ***
   *

Jelly, 46 bytes (non-competing)

“3ḅaė;œ»
Ɠ!
ÆRS
ɠQḲŒl
4Ḷ¤‘+¤ṖṚṭ×”*Fµ4Ḷṭ4ḶṚṖ¤F×⁶+⁸Y

Try it online! (Copy/paste each snippet)

Apparently, the restriction did not restrict golfing :) Just suggestions for the last one, of course, please.

NARS2000 APL, 144 bytes (85 characters)

Task 1, 21 bytes (17 characters)

⎕←"Hello, World!"

Task 2, 10 bytes (4 characters)

⎕←!⎕

Task 3, 22 bytes (11 characters)

⎕←+/¯2π⍳2π⎕

Task 4, 53 bytes (29 characters)

⎕←∪Q[26∣Q⍸⍞∩Q←⎕AV[97+⍳26]∪⎕A]

Task 5, 38 bytes (24 characters)

⎕←" *"[1+4<∘.+⍨(⍳3),⊖⍳4]

osascript, 759 Bytes

I knew this was going to be a lot when I started. o-o

Task 1: 15 Bytes

"Hello, World!"

I knew that it was going to be bad from this point.

Task 2: 64 + 8*4 = 96 Bytes

on run a
set o to 1
repeat a
set o to a*o
set a to a-1
end
o
end

Oh gawd.

Task 3: 170 + 8*13 = 274 Bytes

on run a
set o to 0
set t to false
repeat with i from 2 to a
set t to true
repeat with c from 2 to i-1
if i mod c=0 then set t to false
end
if t then set o to o+i
end
end

Dennis ≠ outgolfed.

Task 4: 225 + 8*13 = 329

on run a
set o to""
repeat with i in items of a
repeat with c in characters of i
if c is not in o then
if ASCII number of c<91 then
set o to o&(ASCII character of(ASCII number of c+32))
else
set o to o&c
end
end
end
end
o
end

...

Task 5: 45 Bytes

"   *
  ***
 *****
*******
 *****
  ***
   *"

So, yeah. I knew I was gonna lose from the start. But it was interesting, I'd be interested to know if there's a way to do this in fewer characters. Character map (as provided by Dennis):

   de  h   l  o  r    w     15
a cdef hi  lmnop rstu w    160
a  de        nop rstu       57
abcdef hi  lmnop rstu w    214
                            39
                             0

The character count above is slightly off - newlines made it have issues, as the newlines were uncounted.

NOTE: The reason for not using stuff like a's characters or the like is that the ' character has to be used when executing from the osascript command line. If I had used ', I'd have to use \' or something similar, which wouldn't have helped me at all. Also, it only recognizes " as string capturers, so I was kinda screwed there as well. But that was fun.

CJam, 73 bytes

"Obkkh+'Phukc&"7f^
q~m!
ri){'j3+_3++~},:+
lS-el_&
4{_' *4@-Y*('**+}%_1>W%\+N*

Each line is a full program. Try them online: 1 | 2 | 3 | 4 | 5

Letter map

 bc  f h  k   op    u       18
            m   q            4
        ij       r          17
    e      l      s          7
             n        w y   27

If you want (and each of your programs fits in a line), you can use this CJam program to create a letter map for your own submission.

Pyth, 90 bytes

First attempt...

Task 1: 20 bytes

+"Hello, "+C87"orld!

Task 2, 3 bytes

.!Q

Task 3, 9 bytes

sf}TPTSvz

Task 4, 6+8=14 bytes

@G{rw0

Task 5, 44 bytes

"   *
  ***
 *****
*******
 *****
  ***
   *