1.1.68 अणुदित्सर्वस्य चा अप्रत्यय:

The literal meaning is - अप्रत्यय: अणुदित् सवर्णस्य स्वस्य च. The following definitions are necessary for the sutra.

1. An अप्रत्यय is the de-suffixed part. For example, in इको यणचि, इक् is being de-suffixed  i.e. replaced by the suffix यण्.
2. अण् is a प्रत्याहार
3. उदित् (उत्  इत् ) implies "उत्s as इत् " - i.e. कु,चु,टु,तु,पु (5x5=25 letters)

The sutra implies that the de-suffixed parts and the अण्s would eclipse (or swallow if you will) their सवर्णs. For example, in रमा ईशः or माला छ (मालीयः), the आ is eclipsed.

1.1.67 स्वं रूपं शब्दस्याअशब्दासंज्ञा

Except as शब्दसंज्ञा, the शब्द shows its own स्वरूप (form). The application of prefixes/suffixes that change meaning on a word act upon the meaning/sense/form of the word not on the word itself.

1.1.66 तस्मादित्युत्तरस्य

Just the way तस्मिन् implies सप्तमी, तस्मात् here implies पञ्चमी. In case of पञ्चमी, the effect (whatever task at hand) occurs at the place of उत्तर.

Let's consider the case of लट् for आस् धातु. Here, लट् becomes शानच्. आस् शानच् then becomes आस् शप् शानच् - आस् शानच् (शप्  disappears as लुक्). In पञ्चमी,  ईत्  आदेश applies to आस् शानच् (through ईदास:). Because of the current rule, ईत्  आदेश must apply at आन (of शानच्) resulting in replacement of आ of आन (आदे: परस्य) with ईत्. Hence, आस् शानच् - आस् ईत् न (this leads to आसीन)

Reference for ईदास: needs to be added.

1.1.65 तस्मिन्निति निर्दिष्टे पूर्वस्य

तस्मिन् implies सप्तमी. In case of सप्तमी, the निर्देश occurs at the place of पूर्व. For example, in भूमि उपरि, the इको यणचि applies at the इ after र्, and not उ before प. भूमि उपरि - भूम् य् उपरि - भूम्युपरि.

1.1.64 अलोअन्त्यात् पूर्व उपधा

The once before अलोअन्त्य is called उपधा. The character before the last अल् in a word is called उपधा. For example, in गं, the last अल् is म्. The character before that is अ- which is उपधा.

1.1.63 अचोअन्त्यादि टि

This sutra defines टि group. It says that everything following the last अच् of a word is considered टि. For example, in मनस्, the last अच् is अ after न्. Everything after that, i.e. अस् becomes  टि. Similarly in दधि, the last  अच् is इ. Everything following that is इ itself. Hence, इ becomes टि.

1.1.62 न लुमताङ्गस्य

लुमान् is one word for लुक् श्लु लुप्. The sutra implies that the disappearance of प्रत्यय caused by the three suppresses the लक्षण of प्रत्यय. For example, गर्ग य जस् - गर्ग जस् (लुक् from यञञ्ओश्च).
The लुक् would suppress all that follows from य i.e. गर्ग अस् - गर्गाः follows.

1.1.61 प्रत्ययलोपे प्रत्ययलक्षणम्

Note that the अदर्शन would still keep the character (लक्षण) of a suffix (प्रत्यय). For example, कृत् पित् after ह्रस्वान्त अङ्ग brings तुक् (considering लक्षण of the प्रत्यय even though it has undergone अदर्शन) :

अग्नि चि क्विप् - अग्नि चि क्वि (हलन्त्यं...) - अग्नि चिवि (लशक्वत..) - अग्नि चिव् (उपदेशे अनुनासिक...) - अग्निचि (वेरपृक्तस्य)
तुक् would require कृत् पित् to be before ह्रस्वान्त अङ्ग for it be applicable. अदर्शन of the प्रत्यय doesn't stop the प्रत्यय from being considered for this rule. The current sutra makes तुक् applicable despite the अदर्शन. After तुक् , अग्निचि + सु becomes अग्निचित् etc.

( Reference needs to be added for कृत् पित्).

१.१.५९ प्रत्ययस्यलुक्श्लुलुपः

Still on the subject of अदर्शनं लोपः the sutra describes the three types of अदर्शन - लुक् श्लु लुप्. The instances of अदर्शन are named after respective instances i.e. An अदर्शन caused by लुक् would be considered लुक्based अदर्शन etc.

Taking विशाख as an example,
विशाखायां जातः - विशाखा ङ्इ अण् - विशाखा अण् - विशाखा (४.३.३४*)
Here लुक् occurred for अण् and would occur for टाप् as well.

विशाख - विशाख:

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*4.3.34 श्रविष्ठाफल्गुन्यनुराधास्वातितिष्यपुनर्वसुहस्तविशाखाषाढाबहुलाल् लुक् defines लुक्activity for respective words. फल्गुनी would have ट and श्रविष्ठा would have छण् etc.

फल्गुन्यां जाता - फल्गुनी ट - फल्गुनी अ - फल्गुनी ङ्ईप - फल्गुनी (द्विगु समास as described in 2.1.2.2 )
श्रविष्ठा, अषाढा - छण्

1.1.59 अदर्शनं लोपः

The अनुव्ऋत्ति of this rule is न वेति विभाषा (which makes it appear somewhat out of place). The meaning is simple - a character whose लोप has occurred can disappear.

1.1.58 द्विवचने अचि

The अजादेश caused by द्विवचन is done at the place of स्थानी. The द्वित्व itself can be due to एकाचे द्वे प्रथमस्य. In पपतु:, पा पा अतुस् - results in पा प् अतुस् (अजादेश at स्थानी) and becomes पपतु:.

Note that interpretations of the rule differ in महाभाष्य and वैयाकरणसिद्धान्तकौमुदी. The latter assumes that अजादेश should not occur before द्वित्व is performed. The result (at least in these examples) is not affected because of this difference.

1.1.57 न पदान्तद्विर्वचनवरे यलोपस्वरसवर्णानुस्वारदीर्घजश्चर्विधिषु

The exception rule states that पर्-caused अजादेश is not "applied"in cases when following occur:

1. पदान्त
 अस् तस् - स् तस् - स्त:
If अलोप was स्थानी-based आव् वृद्धि would have occurred. But this being end of पद the स्थानिवत् भाव does not apply.

2. द्विवचन
In the following case, द्वित्व is allowed because यणादेश is not  स्थानिवत् - otherwise अनचि चwould not be applicable.

दध् य त्र - दध् ध् यत्र ( अनचि च )

3. वरे
 या या यङ् - या या य
When above combines with वरच_ then अल्विधि bars स्थानिवत् लोप of अ. The लोप occurs because of वरच_ (due to अच: परस्मिन् पूर्वविधौ) -
या या य + वरच_ - या या य् वर - यायावर:

4. यलोप
 In the following case, if अलोप occurred at स्थानी then कण्डूय्ति would have resulted.

कण्डू यक् - कण्डू य क्तिच् - कण्डू य ति 

5. स्वर
 In the following case, 6.1.187 would have made अक as उदात्त but the current rule bars that and what's उदात्त is ई.

चिकीर्ष अक - चिकीर्षक

6. सवर्ण
 In the following case, स्थानिवत् would have resulted in barring of परसवर्ण due to अनुस्वार. But since there is no स्थानिवत्  due to current rule, the परसवर्ण follows and ण् develops.

शिन् ड_ ढ_इ - शिन् ड_- शिंढि - शिण्ढि

7. अनुस्वार
 स्थानिवत्लोप of अ could have barred अनुस्वार in the following case but the current rule allows अनुस्वार.
शिन् ष् अन्ति - शिंषन्ति

8. दीर्घ
दीर्घ is allowed in the following case because of current rule.
प्रतिदिवन् टा - प्रतिदिव्न् आ - प्रतिदीव्न् आ

9. जश्
In the following case, अच: परस्मिन् पूर्वविधौ would have barred जश्त्व because of झलां जश्झसि. But the current rule let that happen:
घ् धि - ग् धि

10. चर्
खरि च is allowed because of current rule.
अ घ्स् अन् - अ क्स् अन् - अक्षन्