Q:

Moons and messages

This puzzle is part 15 of Gladys' journey across the globe. Each part can be solved independently. Nevertheless, if you are new to the series, feel free to start at the beginning: Introducing Gladys.

Dear Puzzling,  
Today was devoted to the visual arts. I do have to get my fill of museums for the trip! I hope you enjoy today's puzzle. The objective is to place the answers from the first part into the empty slots in the correct order, thereby forming the clues for the second part.  
Wish you were here!
    Love, Gladys.

$a)$ A giant is younger, making unlimited union small out of spite (7)
$b)$ Gesture if I can finally attempt to be important (11)
$c)$ Climb on mysterious monument with no men (5)
$d)$ Bring back some humanoid artistic medium (5)
$e)$ Most of St. Anthony's following in a moment (7)
$f)$ Metal sword wielded by Frodo with no limits (3)
$g)$ Suriname reconstructed ancient language (8)
$h)$ Publication to tide over nation's right wing (7)
$i)$ Tatum and Garfunkel's creative activities (4)
$j)$ Elevated part of body with no head (4)  
1. ____'s moon
  2. ____ frequency
  3. ____ message
  4. ____ other
  5. ____ definition
  6. Millennium ____
  7. ____ city
  8. ____, in the periodic table
  9. Electronic ____
  10. (See 7.)
  11. ____, abbreviated

Gladys will return in "...And they were stumped for a long time".

A:

Standing on the shoulders of several giants, I can gather a complete answer.
Let's first solve the cryptic puzzles:

 a) JUPITER: "a giant": definition part; "younger": JUNIOR, "unlimited union": NIO, "small": S, "S out of spite": PITE, "JUNIOR making NIO PITE": JUPITER (parsing by Egglz and Rubio)
 b) SIGNIFICANT: "gesture": SIGN, "finally attempt": T, "SIGN if I can T" run together; "important": definition part
 c) MOUNT: "climb on": definition part; "monument with no men": MONUT, "mysterious MONUT" (anagram): MOUNT (solved by Rand al'Thor)
 d) RADIO: "some" of "humanoid artistic": OIDAR, "bring back OIDAR" (reverse): RADIO; "medium": definition part
 e) INSTANT: "Most of St. Anthony's": STANT, "STANT following in": INSTANT; "a moment": definition part (solved by Rand al'Thor)
 f) TIN: "metal": defintion part; "sword wielded by Frodo": STING, "STING with no limits": TIN
 g) SUMERIAN: "Suriname reconstructed" (anagram): SUMERIAN; "ancient language": definition part (solved by Rand al'Thor)
 h) EDITION: "publication": defintion part; "tide over" (reverse): EDIT,  "nation's right wing": ION (solved by Rand al'Thor)
 i) ARTS: "Tatum and Garfunkel": ARTS, referring to Art Tatum and Art Garfunkel); "creative activities": definition part (solution posted by logically complete, explanation hinted by Strawberry and posted by Omega Krypton)
 j) HIGH: "elevated": definition part; "part of body": THIGH, "THIGH with no head": HIGH (solved by Omega Krypton)

Then fill the blanks of the non-cryptic puzzles with these, and solve them:

 01. IO - JUPITER's moon (a)
 02. RF - RADIO frequency (d)
 03. IM - INSTANT message (e) (by Omega Krypton)
 04. SO - SIGNIFICANT other (b)
 05. HD - HIGH definition (j)
 06. ME - Millennium EDITION (h) (by Rand al'Thor)
 07. UR - SUMERIAN city (g) (by Rand al'Thor)
 08. SN - TIN, in the periodic table (f)
 09. EA - Electronic ARTS (i)
 10. UR - again
 11. MT - MOUNT, abbreviated (c) (by Omega Krypton and/or Rand al'Thor)  

Finally her location is revealed by

 taking first the 11 letters of the non-cryptic solutions, then all the second letters (and adding some spaces to taste)

to be the

 Irish Museum of Modern Art

A:

Cryptic Clues Part
a) A giant is younger, making unlimited union small out of spite (7)

 JUPITER (junior meaning younger, without 'nio' from unlimited union and adding the 'pit' from 'spite') by @Egglz

b) Gesture if I can finally attempt to be important (11)

 SIGNIFICANT, by @pt314 (Gesture => "sign" + "if i can" + attemp"t" (finally), def = important) 

c) Climb on mysterious monument with no men (5)

 MOUNT, by @RandAlThor

d) Bring back some humanoid artistic medium (5)

e) Most of St. Anthony's following in a moment (7)

 INSTANT, by @RandAlThor

f) Metal sword wielded by Frodo with no limits (3)

 ROD (Frodo - f - o (limits) , def = metal sword)

g) Suriname reconstructed ancient language (8)

 SUMERIAN, by @RandAlThor

h) Publication to tide over nation's right wing (7)

 EDITION, by @RandAlThor

i) Tatum and Garfunkel's creative activities (4)

 ARTS, by @logicallyincomplete (Art Tatum, Art Garfunkel => two "Art's" , def = creative activities)

j) Elevated part of body with no head (4)

 HIGH (Thigh - h, def = elevated) [Sidenote: for so long i thought it would end in -ed]

Crossword Clues Part
Foreword

 Seems like each of them yields two letters...

____'s moon

 IO, blank is Jupiter from (a), by @Egglz

____ frequency

 HF, blank is high from (j)

____ message

 IM, blank is instant from (j)

____ other

 SO, blank is significant from (b), by @pt314

____ definition

Millennium ____

____ city

 UR, blank is Sumerian from (g)

____, in the periodic table

Electronic ____

 ??, blank is rod from (j)

(See 7.)

 UR, blank is Sumerian from (g)

____, abbreviated

 MT, blank is Mount from (c)

A:

Solving clues
a) A giant is younger, making unlimited union small out of spite (7) -- unsolved

 "unlimited union" probably means NIO, and I'm guessing "a giant" is the def part.

b) Gesture if I can finally attempt to be important (11) -- unsolved

 Probably "gesture" is the def part, and maybe "finally attempt" means something like EMPT.

c) Climb on mysterious monument with no men (5)

 MOUNT (climb, I guess "mysterious" is an anagram indicator, monument with no "men")

e) Most of St. Anthony's following in a moment (7)

 INSTANT (most of St Anthony's, following in, a moment)

f) Metal sword wielded by Frodo with no limits (3)

 ROD (metal sword, Frodo with no limits)

g) Suriname reconstructed ancient language (8)

 SUMERIAN (ancient language, anagram of Suriname)

h) Publication to tide over nation's right wing (7)

 EDITION (publication, tide backwards and right wing of nation)

i) Tatum and Garfunkel's creative activities (4) - unsolved

 This probably needs to be solved by someone with some knowledge of popular culture.

j) Elevated part of body with no head (4) - solved by Omega Krypton

 HIGH (elevated, thigh with no head)

Placing in order

 INSTANT message --> SMS

 Millennium EDITION?

 SUMERIAN city --> UR

 Electronic ROD?

 MOUNT, abbreviated? -> MT

