Gathering for fraternities and sororities

Question

_{This puzzle is part of the Monthly Topic Challenge #4: Cross-*non*-words}

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I've already got two crosswords for the challenge.
Recently, the fraternities, the sororities and I worked together to make a third crossword in memory of their naming convention.

Solve each cryptic clue, then transliterate the solution words into another language. Fill in the transliterated words in the proper position in the grid. The letter in parentheses would describe my lucky number^{[citation needed]}.

The three "hidden" clues are to be solved with the help of the language's history.

Notes to the grid: The "order of magnitude" is omitted in the grid. The 2 on the second line is supposed to have a 9 overlapping it in the same cell, thus the digits from 1 to 9 would each appear exactly three times in the grid - once for "x10⁰" group, once for "x10¹" group, and once for "x10²" group.
The image version is not yet available.

# # # # # # # # # # # # # # # # # # 6 
# # # # # # # # # # # # # # 2 . 4 . . 
# # # # # # # # # # # # # # . # . # # 
# # # # # # # # # # # # # # . # # # # 
# # # # # # # # # # # # 7 # . # # # # 
# # # # # # # # # # # # 8 . . # # # # 
# # # # # # # # # 2 # # . # # # # # # 
# # # # 4 # # # # 1 . 5 . # # # # # # 
# # # # . # 3 # # . # . # # # # # # # 
# # # # . # . # # 8 . # # # # # # # # 
# # # # 3 . . . 6 . # # 5 # # # # # # 
# # # # . # . # . # # # 7 . # # # # # 
# # # # . # . # . # # 6 . # # # # # # 
# # # # # # # # . # # # . # # # # # # 
# # # # # 8 # 7 . . 9 . . . # # # # # 
# # # # # . # # . # . # . # # # # # # 
# # # # 4 . . . . #(.)# # # # # # # # 
# # # 9 # . # # # # . # # # # # # # # 
# # 2 . . . # # # # . # # # # # # # # 
# 1 # . # # # # # # # # # # # # # # # 
1 . 3 . # # # # # # # # # # # # # # # 
# # . # # # # # # # # # # # # # # # # 
# 5 . # # # # # # # # # # # # # # # # 

Group 1 (x 10⁰)

1. Testing version mixing up asphalt (removed - st)
2. Guess a public testing version
3. Light particle start guessing before noon, forth and back
4. Triangle appears with the dot, top of the asymptote
5. Quantum particle and small particle long before ending
6. (Unknown)
7. Organize Tale hiding special function
8. Efficiency of work describes when they could reach the goal
9. The volunteers's any angle

Group 2 (x10¹)

1. Unfinished note: Thank you a bit
2. Sportswear coefficient
3. Animal lawyer wavelength
4. Maximum extreme voltage is one in a million
5. Unusually hidden radio frequency
6. Random variable of president who recently expanded this sequence [Down]
7. Zero-length unit virus: newest discovery
8. Euler's constant reversing intellectual property
9. (Unknown)

Group 3 (x10²)

1. Take bad lady's density
2. Sum shame minus time
3. Twice Euler's constant is half of zodiac sign
4. Up is inverted left on meson
5. Part of philosophy's golden ratio
6. Communist leader doesn't finish one test of statistics [Across]
7. Corrupted ISP particle
8. Any angle over time in some game
9. (Unknown)

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Task 1: As you can guess...

The main task is to solve the crossword.
It is recommended that you use lowercase letters in the grid so that the crossword can be qualified for the challenge.
All diacritics, except the acute accent^{[citation needed]}, are ignored.

Task 2: (optional)

Try to make another crossword grid using these 27 words you wrote in my grid, so that if a blank version of your grid (without clue numbers) and a list of the 27 words are given, one can find a unique method to locate each word in your grid, similar to this.
Is it actually possible?

Hint 1:

The "order of magnitude" is related to the old numerals in the language.

Answer 1

Your lucky number is

3.14159... or π

The fraternities and sororities suggest

we will be using Greek letters.

In fact each clue

clues a Greek letter. The three "(Unknown)" clues clue three old Greek letters that fell out of use. Two are them are digamma and koppa. Thanks to @Bart-Jan van Rossum for finding the third: sampi.

Group 1 (x 10^0)

ALPHA 1. Testing version mixing up asphalt (removed - st)
BETA 2. Guess a public testing version
GAMMA 3. Light particle start guessing before noon, forth and back
DELTA 4. Triangle appears with the dot, top of the asymptote
EPSILON 5. Quantum particle and small particle long before ending
DIGAMMA 6. (Unknown)
ZETA 7. Organize Tale hiding special function
ETA 8. Efficiency of work describes when they could reach the goal
THETA 9. The volunteers's any angle

Group 2 (x10^1)

IOTA 1. Unfinished note: Thank you a bit
KAPPA 2. Sportswear coefficient
LAMBDA 3. Animal lawyer wavelength
MU 4. Maximum extreme voltage is one in a million
NU 5. Unusually hidden radio frequency
XI 6. Random variable of president who recently expanded this sequence [Down]
OMICRON 7. Zero-length unit virus: newest discovery
PI 8. Euler's constant reversing intellectual property
SAMPI 9. (Unknown)

Group 3 (x10^2)

RHO 1. Take bad lady's density
SIGMA 2. Sum shame minus time
TAU 3. Twice Euler's constant is half of zodiac sign
UPSILON 4. Up is inverted left on meson
PHI 5. Part of philosophy's golden ratio
CHI 6. Communist leader doesn't finish one test of statistics [Across]
PSI 7. Corrupted ISP particle
OMEGA 8. Any angle over time in some game
KOPPA 9. (Unknown)

The filled in grid

# # # # # # # # # # # # # # # # # # ξ 
# # # # # # # # # # # # # # σ α μ π ι 
# # # # # # # # # # # # # # ί # υ # # 
# # # # # # # # # # # # # # γ # # # # 
# # # # # # # # # # # # ζ # μ # # # # 
# # # # # # # # # # # # ή τ α # # # # 
# # # # # # # # # κ # # τ # # # # # # 
# # # # ύ # # # # ά λ φ α # # # # # # 
# # # # ψ # γ # # π # ι # # # # # # # 
# # # # ι # ά # # π ι # # # # # # # # 
# # # # λ ά μ β δ α # # έ # # # # # # 
# # # # ο # μ # ί # # # ψ ι # # # # # 
# # # # ν # α # γ # # χ ι # # # # # # 
# # # # # # # # α # # # λ # # # # # # 
# # # # # ω # ό μ ι κ ρ ο ν # # # # # 
# # # # # μ # # μ # ό # ν # # # # # # 
# # # # δ έ λ τ α #(π)# # # # # # # # 
# # # θ # γ # # # # π # # # # # # # # 
# # β ή τ α # # # # α # # # # # # # # 
# ρ # τ # # # # # # # # # # # # # # # 
ι ω τ α # # # # # # # # # # # # # # # 
# # α # # # # # # # # # # # # # # # # 
# ν υ # # # # # # # # # # # # # # # # 

For task 2

It is impossible. ξι and χι both start with a letter that do not appear in the names of any other letter so they only cross with the iota. It would be impossible to disambiguate which belongs where.