1. Where has the सूत्रम् 2-2-28 तेन सहेति तुल्ययोगे (used in step 2) been used in verses 5-10 of Chapter Three of the गीता?
Answer: The सूत्रम् 2-2-28 तेन सहेति तुल्ययोगे has been used in the form सहयज्ञाः (प्रातिपदिकम् ’सहयज्ञ’, स्त्रीलिङ्गे द्वितीया-बहुवचनम्) in the following verse of the गीता –

सहयज्ञाः प्रजाः सृष्ट्वा पुरोवाच प्रजापतिः ।
अनेन प्रसविष्यध्वमेष वोऽस्त्विष्टकामधुक् ॥ ३-१०॥

यज्ञेन सह (वर्तन्ते) = सहयज्ञाः (प्रजाः) – (People) along with sacrifice

The derivation of the compound प्रातिपदिकम् ’सहयज्ञ’ is similar to the derivation of the compound प्रातिपदिकम् ’सहबल’ as shown in the post.
The gender of a बहुव्रीहि: compound matches that of which it qualifies. In the present example सहयज्ञाः is qualifying the feminine noun प्रजाः। Hence we assign the feminine gender to the compound प्रातिपदिकम् ‘सहयज्ञ’। Since this प्रातिपदिकम् is used here in the feminine gender we have to add the appropriate feminine affix (स्त्रीप्रत्यय:)।
= सहयज्ञ + टाप् । By 4-1-4 अजाद्यतष्टाप्‌।
= सहयज्ञ + आ । अनुबन्ध-लोपः by 1-3-3 हलन्त्यम्, 1-3-7 चुटू, 1-3-9 तस्य लोपः।
= सहयज्ञा । By 6-1-101 अकः सवर्णे दीर्घः। It declines like सीता-शब्द:। द्वितीया-बहुवचनम् is सहयज्ञाः।

2. Commenting on the सूत्रम् 2-2-28 तेन सहेति तुल्ययोगे the सिद्धान्तकौमुदी says – तुल्ययोगवचनं प्रायिकम्‌। सकर्मक:। Please explain.
Answer: The requirement that – in order for the सूत्रम् 2-2-28 to apply सह should denote ‘equal connection (with an action)’ – is प्रायिकम् meaning that it usually applies but is not universal. We infer this from the fact that पाणिनि: himself has used compounds like ‘सकर्मक’ (ref. 1-3-53 उदश्चरः सकर्मकात्) in the अष्टाध्यायी। In the compound ‘सकर्मक’ the term सह does not denote ‘equal connection (with an action.)’ It simply denotes ‘existing.’ Hence सकर्मक: means विद्यमानकर्मक: – one (the verbal root) whose object exists. Thus it means a transitive verbal root.
Note: In the compound सकर्मक: the (optional) substitution ‘स’ (in place of सह) is as per the सूत्रम् 6-3-82 वोपसर्जनस्य। The affix ‘क’ at the end of ‘सकर्मक’ is a तद्धित: affix added स्वार्थे (without changing the meaning of the compound.)

3. What is the विग्रह: of the compound जातहर्ष: used in the verses?
Answer: The विग्रह: of the compound जातहर्ष: is –
जातो हर्षो यस्य/यस्मिन् सः = जातहर्ष: (श्रीकृष्णः) – One (Śrī Kṛṣṇa) whose (or in whom) joy/delight has arisen.

The derivation of the compound प्रातिपदिकम् ’जातहर्ष’ is similar to the derivation of the compound प्रातिपदिकम् ’कृतकृत्य’ as shown in the following post – https://avg-sanskrit.org/2015/11/19/कृतकृत्यम्-mas

The gender of a बहुव्रीहि: compound matches that of which it qualifies. In the present example ‘जातहर्ष’ is qualifying Śrī Kṛṣṇa. Hence we assign the masculine gender to the प्रातिपदिकम् ‘जातहर्ष’। It declines like राम-शब्द:। प्रथमा-एकवचनम् is जातहर्ष:।

4. In which word in the verses has the affix ‘क’ been used?
Answer: The affix ‘क’ has been used in the form प्रतपत्रम् (प्रातिपदिकम् ’प्रतपत्र’, नपुंसकलिङ्गे द्वितीया-एकवचनम्)।

प्रतपात् (आतपात्) त्रायत इति प्रतपत्रम् – an umbrella (which protects from heat.)

’त्र’ is a कृदन्त-प्रातिपदिकम् (participle form) derived from the verbal root √त्रै (त्रैङ् पालने १. ११२०).

The (compound) प्रातिपदिकम् ’प्रतपत्र’ is derived as follows:
प्रतप + ङसिँ + त्रा + क । By 3-2-4 सुपि स्थः – This rule is split into two parts. The first part is सुपि – When in composition with a पदम् which ends in a सुप् affix (ref. 4-1-2 स्वौजसमौट्छष्टा..), a verbal root which ends in the letter ‘आ’ may take the affix ’क’। (We don’t need the second part of the सूत्रम् in the present example.)
Note: As per the सूत्रम् 6-1-45 आदेच उपदेशेऽशिति, The ending ‘ऐ’ of the verbal root √त्रै is replaced by the letter ‘आ’।
Note: In the सूत्रम् 3-2-4 (सुपि), the term सुपि ends in the seventh (locative) case. Hence ’प्रतप + ङसिँ’ gets the उपपद-सञ्ज्ञा here by 3-1-92 तत्रोपपदं सप्तमीस्थम्‌।
= प्रतप + ङसिँ + त्रा + अ । अनुबन्ध-लोप: by 1-3-8 लशक्वतद्धिते and 1-3-9 तस्य लोपः।
= प्रतप + ङसिँ + त्र् + अ । By 6-4-64 आतो लोप इटि च।
= प्रतप + ङसिँ + त्र । We form a compound between ’प्रतप + ङसिँ’ (which is the उपपदम्) and ’त्र’ by using the सूत्रम् 2-2-19 उपपदमतिङ्।
In the compound, ’प्रतप + ङसिँ’ is placed in the prior position as per 2-2-30 उपसर्जनं पूर्वम्‌। Note: Here ’प्रतप + ङसिँ’ is the उपपदम् and hence it gets the उपसर्जन-सञ्ज्ञा by 1-2-43 प्रथमानिर्दिष्टं समास उपसर्जनम् because in the सूत्रम् 2-2-19 उपपदमतिङ् (which prescribes the compounding) the term उपपदम् ends in the nominative case.
’प्रतप + ङसिँ + त्र’ gets प्रातिपदिक-सञ्ज्ञा by 1-2-46 कृत्तद्धितसमासाश्च। This allows 2-4-71 to apply in the following step.
= प्रतपत्र । By 2-4-71 सुपो धातुप्रातिपदिकयोः।

5. Which वार्तिकम् justifies the use of a third case affix in the form कुसुमतया used in the commentary?
Answer: The third case affix occurring in the form कुसुमतया (स्त्रीलिङ्ग-प्रातिपदिकम् ‘कुसुमता’, तृतीया-एकवचनम्) is justified by the वार्तिकम् (under 2-3-18 कर्तृकरणयोस्तृतीया) प्रकृत्यादिभ्य उपसङ्ख्यानम्– A third case affix (‘टा’, ‘भ्याम्’, ‘भिस्’) is used following the प्रातिपदिकानि (nominal stems) ‘प्रकृति’ etc.
Note: आकृति-गणोऽयम्। The list ‘प्रकृति’ etc is a आकृति-गण: – which is a class or group of words in which some words are actually mentioned and room is left to include others which are found undergoing the same operations. So even though ‘कुसुमता’ is not specifically listed in the प्रकृत्यादि-गण:, from the usage we have to understand it to be included there.

कुसुमतया is an adverb to the action कल्पितः ‘imagined.’ कुसुमतया कल्पितः (तुषार:) = (Mist/spray) imagined in the manner of a flower. Hence it means – (Mist/spray) imagined to be flowers.

6. How would you say this in Sanskrit?
“Rāvaṇa along with Mārica went to the Daṇḍaka forest to abduct Sītā.” Construct a बहुव्रीहि: compound for ‘along with Mārica’ (Rāvaṇa went) = मारीचेन सह (रावणो जगाम)। Use the verbal root √हृ (हृञ् हरणे १. १०४६) preceded by the उपसर्ग: ‘अप’ for ‘to abduct.’
Answer: सहमारीचः रावणः सीताम् अपहर्तुम् दण्डकारण्यम् जगाम = सहमारीचो रावणः सीतामपहर्तुं दण्डकारण्यं जगाम।

Easy questions:
1. In which word in the verses has the सूत्रम् 7-1-78 नाभ्यस्ताच्छतुः been used?
Answer: The सूत्रम् 7-1-78 नाभ्यस्ताच्छतुः has been used in the form विदधत् (प्रातिपदिकम् ’विदधत्’, पुंलिङ्गे प्रथम-एकवचनम्)।
’विदधत्’ is शतृँ-प्रत्ययान्त-शब्दः derived the verbal root √धा (डुधाञ् धारणपोषणयोः | दान इत्यप्येके ३. ११) with the उपसर्ग: ‘वि’। Since this verbal root is in the जुहोत्यादि-गणः, the प्रातिपदिकम् ’विदधत्’ is derived through reduplication (अभ्यासः)। There is no augment ‘नुँम्’ (by 7-1-70 उगिदचां सर्वनामस्थानेऽधातोः) in the form विदधत् (प्रातिपदिकम् ‘विदधत्’, पुंलिङ्गे प्रथमा-एकवचनम्) because of the निषेध-सूत्रम् 7-1-78 न अभ्यस्ताच् छतुः – the ‘शतृँ’ affix that follows an अभ्यस्तम् (ref 6-1-5 उभे अभ्यस्तम्) does not get the नुँम् augment.

Please see answer to question 3 in the following comment for derivation of the form विदधत् from the प्रातिपदिकम् ‘विदधत्’ – https://avg-sanskrit.org/2011/12/11/जगाद-3as-लिँट्/#comment-2944

2. Where has the सूत्रम् 2-4-72 अदिप्रभृतिभ्यः शपः been used in the commentary?
Answer: The सूत्रम् 2-4-72 अदिप्रभृतिभ्यः शपः has been used in the form याति – derived from the verbal root √या (या प्रापणे, अदादि-गणः, धातु-पाठः २. ४४).

Please see answer to easy question 2 in the following comment for the derivation of the form याति – http://avg-sanskrit.org/2012/08/16/महीयते-3as-लँट्/#comment-4258