1. Which other compounds (besides बहुसवम्) used in the verses are बहुव्रीहि: compounds?
Answer: The compounds महात्मा (प्रातिपदिकम् ‘महात्मन्’, पुंलिङ्गे प्रथमा-एकवचनम्) and अविचलेन्द्रियः (प्रातिपदिकम् ‘अविचलेन्द्रिय’, पुंलिङ्गे प्रथमा-एकवचनम्) are also बहुव्रीहि: compounds.

The लौकिक-विग्रहः is as follows –
महानात्मा यस्य सः = महात्मा – high-souled
अविचलानीन्द्रियाणि यस्य सः = अविचलेन्द्रियः – one whose senses are not perturbed.

The derivation of the compound प्रातिपदिकम् ‘महात्मन्’ is similar to the derivation of the compound प्रातिपदिकम् ‘महाबाहु’ as shown in answer to question 1 in the following comment – https://avg-sanskrit.org/2015/10/29/दत्तराज्यः-mns/#comment-35681. Except that we need to apply 6-1-101 अकः सवर्णे दीर्घः a second time to get
महा + आत्मन् = महात्मन् as the final compound प्रातिपदिकम्।

The derivation of the compound प्रातिपदिकम् ‘अविचलेन्द्रिय’ is similar to the derivation of the compound प्रातिपदिकम् ‘चतुर्भुज’ as shown in the following post – https://avg-sanskrit.org/2015/10/22/चतुर्भुजः-mns/#comment-35673. Except that after we need to apply 6-1-87 आद्‍गुणः at the end to get
अविचल + इन्द्रिय = अविचलेन्द्रिय as the final compound प्रातिपदिकम्।
Note: ‘अविचल’ itself is a नञ्-तत्पुरुषः compound. Please see the derivation of the compound प्रातिपदिकम् ‘अविचल’ in answer to question 5 below.

2. Can you spot a बहुव्रीहि: compound in verses 10-16 of Chapter Five of the गीता?
Answer: The compound नवद्वारे (प्रातिपदिकम् ‘नवद्वार’, नपुंसकलिङ्गे सप्तमी-एकवचनम्) in the following verse of Chapter Five of the गीता is a बहुव्रीहि: compound.

नव द्वाराणि यस्य तत् = नवद्वारम् – city/body (one which has nine doors/openings)

सर्वकर्माणि मनसा संन्यस्यास्ते सुखं वशी |
नवद्वारे पुरे देही नैव कुर्वन्न कारयन्‌ || 5-13||

अलौकिक-विग्रह: –
नवन् जस् + द्वार जस् । By 2-2-24 अनेकमन्यपदार्थे।
By 1-2-43 प्रथमानिर्दिष्टं समास उपसर्जनम् – the term ‘नवन् जस्’ as well as ‘द्वार जस्’ gets the designation उपसर्जनम् because in the सूत्रम् 2-2-24 अनेकमन्यपदार्थे (which prescribes the compounding) the term अनेकम् ends in the nominative case. And hence 2-2-30 cannot help us in deciding which term to place in the prior position in the compound.
As per 2-2-35 सप्तमीविशेषणे बहुव्रीहौ the adjective ‘नवन् जस्’ is placed in the prior position in the compound.
Note: ‘नवन् जस् + द्वार जस्’ gets the designation प्रातिपदिकम्‌ by 1-2-46 कृत्तद्धितसमासाश्च। This allows 2-4-71 to apply in the next step.
= नवन् + द्वार । By 2-4-71 सुपो धातुप्रातिपदिकयोः।
Note: Here ‘नवन्’ has the designation पदम् as per 1-4-14 सुप्तिङन्तं पदम् with the help of 1-1-62 प्रत्ययलोपे प्रत्ययलक्षणम्। This allows 8-2-7 नलोपः प्रातिपदिकान्तस्य to apply in the next step.
= नवद्वार । By 8-2-7 नलोप: प्रातिपदिकान्तस्य।

The gender of a बहुव्रीहि: compound matches that of which it qualifies. In the present example the neuter noun ‘पुर’ is being qualified. Hence we assign the neuter gender to the compound प्रातिपदिकम् ‘नवद्वार’। It declines like वन-शब्द:।

3. Which सूत्रम् justifies the use of a second case affix in the form बहुसवम्?
Answer: The use of a second case affix in the word बहुसवम् is justified by the सूत्रम् 2-3-5 कालाध्वनोरत्यन्तसंयोगे – A second case affix (‘अम्’, ‘औट्’, ‘शस्’) is used following a प्रातिपदिकम् (nominal stem) denoting (measure of) time or (measure of) path/road when continuous connection (with a quality or action or thing) is meant.

In the present example, ‘बहुसव’ (many years) denotes a measure of time which has continuous connection with the action of ‘having spent’ denoted by the the verbal root √नी (णीञ् प्रापणे १. १०४९) present in the word नीत्वा। Hence the प्रातिपदिकम् ‘बहुसव’ takes a second case affix as per the सूत्रम् 2-3-5.

4. Where has the सूत्रम् 2-3-13 चतुर्थी सम्प्रदाने been used in the verses?
Answer: The सूत्रम् 2-3-13 चतुर्थी सम्प्रदाने has been used to prescribe the fourth case affix in the form पुत्राय (पुंलिङ्ग-प्रातिपदिकम् ‘पुत्र’, चतुर्थी-एकवचनम्)।

As per the सूत्रम् 2-3-13 चतुर्थी सम्प्रदाने – A fourth case affix (‘ङे’, ‘भ्याम्’, ‘भ्यस्’) is used to denote the सम्‍प्रदानम् (recipient) provided it has not been expressed otherwise.

The सूत्रम् 2-3-13 applies here because ‘पुत्र’ (‘son’) gets the designation सम्‍प्रदानम् (recipient) by the सूत्रम् 1-4-32 कर्मणा यमभिप्रैति स सम्प्रदानम् – That कारकम् (participant in the action) whom the doer wishes to connect/endow with the object of an action is called सम्‍प्रदानम् (recipient.)

The doer is महात्मा (Dhruva), the object of the action (अदात्) is नृपासनम् (throne), and the सम्‍प्रदानम् (recipient) is ‘पुत्र’। Therefore the प्रातिपदिकम् ‘पुत्र’ takes a fourth case affix by the सूत्रम् 2-3-13 चतुर्थी सम्प्रदाने।

5. How many times has the negation particle नञ् been used in the verses?
Answer: The negation particle नञ् has been used in the following forms – प्रातिपदिकम् ‘अभोग’ (used in अभोगैः), प्रातिपदिकम् ‘अशुभ’ (used in अशुभक्षयम्) and प्रातिपदिकम् ‘अविचल’ (used in अविचलेन्द्रियः)।

The लौकिक-विग्रहः is as follows –
न भोगः = अभोगः – self-denial (absence of pleasure)
न शुभम् = अशुभम् – evil (not auspicious)
न विचलम् = अविचलम् – not perturbed

In the form अभोगः the negation particle नञ्‌ conveys the sense of अभावः – absence of that which is being negated. In the other two forms – अशुभम् and अविचलम् – the negation particle नञ्‌ conveys the sense of विरोध: – opposite of that which is being negated.

The derivation of the compound प्रातिपदिकम् ‘अभोग’/’अशुभ’/‘अविचल’ is similar to the derivation of the compound प्रातिपदिकम् ‘अपर्याप्त’ shown in answer to question 1 in the following comment – https://avg-sanskrit.org/2015/08/10/अब्राह्मणः-mns/#comment-35496

6. How would you say this in Sanskrit?
“I lived in India for a period of many years.”
Answer: (अहम्) भारतदेशे बहुसवम् कालम् न्यवसम् = भारतदेशे बहुसवं कालं न्यवसम्।

Easy questions:
1. In which word in the verses has लिँट् been used?
Answer: लिँट् has been used in the form शशास – derived from the verbal root √शास् (शासुँ अनुशिष्टौ २. ७०).

The विवक्षा is लिँट्, कर्तरि प्रयोग:, प्रथम-पुरुषः, एकवचनम्।
शास् + लिँट् । By 3-2-115 परोक्षे लिँट् – The affix लिँट् (Perfect Tense) comes after a verbal root in the sense of the past not of today, provided that the action is unperceived by the narrator.
= शास् + ल् । अनुबन्ध-लोपः by 1-3-2 उपदेशेऽजनुनासिक इत्, 1-3-3 हलन्त्यम्, 1-3-9 तस्य लोपः।
= शास् + तिप् । By 3-4-78 तिप्तस्झिसिप्थस्थमिब्वस्मस्तातांझथासाथांध्वमिड्वहिमहिङ्, 1-3-78 शेषात् कर्तरि परस्मैपदम्।
= शास् + णल् । By 3-4-82 परस्मैपदानां णलतुसुस्थलथुसणल्वमाः, 1-3-10 यथासंख्यमनुदेशः समानाम्।
= शास् + अ । अनुबन्ध-लोपः by 1-3-3 हलन्त्यम्, 1-3-7 चुटू, 1-3-9 तस्य लोपः।
= शास् शास् + अ । By 6-1-8 लिटि धातोरनभ्यासस्य।
= शा शास् + अ । By 7-4-60 हलादिः शेषः।
= शशास । By 7-4-59 ह्रस्वः।

2. Which सूत्रम् prescribes the लुक् elision of the affix सिँच् in the form अदात्?
Answer: The सूत्रम् 2-4-77 गातिस्थाघुपाभूभ्यः सिचः परस्मैपदेषु prescribes the लुक् elision of the affix सिँच् in the form अदात् – derived from the verbal root √दा (डुदाञ् दाने ३. १०).

Please see the following post for derivation of the form अदात् – http://avg-sanskrit.org/2012/04/26/अदात्