knowledge - 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.}

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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

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

Answer

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