There's a topic with many of the codes here.

Not sure if it has exactly what you're looking for, but it might...

You seem to be asking "can someone code the shell part of my shell ?"
Also, your questions aren't specific enough: "running a user defined program" is different if the prog is archived or not; displaying a sprite is easy but it depends of where the sprite is stored.

Something you could do is learn ASM to get the hex codes yourself (you won't even need hex codes then ) or learn Axe, that could do the job too. Else, I don't think there are hex codes that would fit your needs and not create any bugs (like running an archived prog like an unarchived prog, or run an ASM prog like a basic prog would crash I guess).

Also, you might want to read this.

I would do that but I don't want to use any external libs. that's the big mistake I made in my last version of Fittle. It was a mistake because most people get that support from doors cs.

Now, I hate to kind of be a jerk like this, but the routines you asked for are actually each major undertakings themselves. These aren't really things you just whip up. But, you are in luck because I have already written one of the more difficult ones.

Grouping/Ungrouping - This one sucked to make, but I actually did do it a long time ago. final version
Usage:
 Str3 = group name (all uppercase of course, 8 letter max) Ex: MYGROUP
 Str4 = list of programs to group with identifiers Ex: +A+B-IMPOSBLE-GROUP2
   Identifiers:
     + = regular program (you can edit it in the editor)
     - = protected program (you can't edit it in the editor)
 I'm not positive if it will work with the wrong identifier, never tried it.

Hex codes - Rather than giving you the hex codes, this program will spit out the hex code for a program name given in Str1. link
Usage:
  Str1 = name of program to unsquish with identifier padded to be exactly 8 characters with spaces (from above) Ex: -GROUP2__
  It will change the first letter of the program to an A and fill it with hex.
  To copy/paste the hex codes, just use RCL to recall the program. Ex: RCL prgmAROUP2

  However, turning already written assembly code back into hex is simply doubling the size.

Displaying Sprites - I'm sure you can find a program on ticalc.org.
Program list - That wouldn't be all that hard to make, but again, ticalc.org can probably help
Running a program - This is a serious undertaking. If you want to run simple unarchived basic programs and homescreen assembly programs, then I suppose you could probably find a 100 byte routine on ticalc.org. But if you want to start running ion, mirage, and dcs programs, then you are going to either need an app, or some serious hacking skills. Without hacking, (since the hacking would be unreliable), you are going to need to include all of the Mirage routines inside this subroutine and put them in the proper place. This really cannot be done without a shell. (zStart has an entry point for running programs, jussayin')
Getting program details - again, ticalc.org.

But honestly, like everyone has been saying, this is something you want to do in at least Axe. And really, axe isn't even designed for this kind of thing, asm is really the language that should be used for shells. If you are going to use BASIC, I would recommend you use the Celtic3 libs because they are included in DCS and requiring DCS isn't really the end of the world.

I guess I should have detailed my post more...
CopyProg will copy any variable from memory to a var in RAM. So you can copy Appvars or strings to a program variable and run the program (so you can actually have the user input BASIC code in a string and execute it). It also lets you delete vars.
GetName will return names of variables in alphabetical order. The output string can be used by CopyProg, too. So you can input a list saying "grab the name of the nth variable of this type." So you can make a menu for the user to select from appvars, programs, strings, et cetera, then copy them to another var to run them. Plus, it returns info about the size of the var and archive status.

At this point, those two programs together are a little over 360 bytes. That is tiny for so much functionality.

CopyProg 2.00 Combines the functions of both (using the same inputs) and also gives line reading support and line deleting. and a few other functions. This is over 600 bytes.

If you need other programs, I also have:
-A program that lets you execute a string of hexadecimal
-A program to execute a string as code
-A program to convert a list to a byte string (in case your list is integers 0 to 255) and it does the reverse (string to list). This is great for compression and is fast
-A flag editor so that you can edit internal flags (sometimes useful)
-A program to convert a hex string to pure data
-A program to convert a program to a hex string
-A program to return the number of chars in a string (so "sin(cos(37))" would return 12)
-A program to return the width of a string in the small font

Most of those are all around 100 bytes or less

Xeda how did you learn the hex codes? I think that I might try to do that depending on how long it would take to learn the basics of it.

It is the same than learning ASM just that you replace the commands by their hex equivalents (like "ret"==C9 or "xor a"==AF)