Is a sinful/bad unintentional thought a sin?

1) Amrita Bindu Upanishad (अमृतबिन्दु) says mind is the cause of bondage as well liberation as below:

मन एव मनुष्याणां कारणं बन्धमोक्षयोः ।
बन्धाय विषयासक्तं मुक्त्यै निर्विषयं स्मृतम् ॥ २॥
mana eva manushyānām kāranam bandha-mokshayoh ।
bandhaya visayāsakatam muktyai nirvisayam smrutam ॥2॥

Verse-2: It is indeed the mind that is the cause of men's bondage and liberation. The mind that is attached to sense-objects leads to bondage, while dissociated from sense-objects it tends to lead to liberation. So they think.

Any bad thoughts appearing in the mind should be abandoned because only those thoughts will lead to deeds ultimately causing bondage.

2) Paṭṭiṉattār (பட்டினத்தார்), a 14th century Tamil Shaivite, one of the most popular Saint, Jnāni and Siddhar says committing sin in mind is also a sin (or dosha/दोषः) as below:

சொல்லால் வருங்குற்றஞ் சிந்தையால் வருந்தோடஞ்செய்த
   பொல்லாத தீவினை பார்வையிற் பாவங்கள் புண்ணியநூல்
அல்லாத கேள்வியைக் கேட்டிடுஞ் தீங்குக ளாயவுமற்று
   எல்லாப்பிழையும் பொறுத்தருள் வாய்கச்சி ஏகம்பனே.

Meaning of the pharse "சிந்தையால் வருந்தோடம்" is sinful thoughts emerged in the mind. Any act, particularly revenge, will be first born in mind as a thought which will be stored as a seed of samskāra (संस्कार/Latent impressions) in Chitta. If the thought (or notion) is nurtured further intensively (teevra/तीव्र), then the seed corresponding to that thought will grow stronger and stronger and someday the act of revenge will be actually committed (or effected) yielding sin (Pāpa/पाप) as karma phalam (कर्म फलं/Fruits of karma) causing not only suffering and unhappiness but also samsāra (संसार/cycle of birth and death).

Any bad thoughts or notions occurring in mind knowingly or unknowingly should never be promoted or nurtured. It is difficult to avoid such thoughts appearing often but one should learn to ignore and never entertain them. If one allow to grow it stronger (seed of samskāra corresponding to sinful thought), then it will emerge often from subconscious mind to the field of conscious mind to trouble.

No, unintentional mental sins do not occur demerit or sin in Kali-yuga (this is applicable only in this yuga).

कलिजुग जोग न जग्य न ग्याना । एक अधार राम गुन गाना॥ सब भरोस तजि जो भज रामहि। प्रेम समेत गाव गुन ग्रामहि॥ सोड भव तर कछु संसय नाहीं। नाम प्रताप प्रगट कलि माहीं॥ कलि कर एक पुनीत प्रतापा। मानस पुन्य होहिं नहिं पापा॥

In the Kali Yuga, there is neither yoga (yogaḥ) nor sacrifice (yajñaḥ), nor even true knowledge (jñānam). The only support (ekamātra-adhāraḥ) is the glorification of Śrī Rāma’s virtues (Śrī-Rāma-guṇa-gānam). Therefore, abandoning all other forms of reliance (sarve bharasāḥ tyaktvā), he who worships Śrī Rāma (yaḥ Śrī-Rāmam bhajati) and sings His virtues (guṇa-grāmam gāyati) with love (prema-sametaḥ).

He alone crosses the ocean of worldly existence (sa eva bhava-sāgaram tarati); there is not the slightest doubt (atra kiñcid api saṁśayaḥ na asti). The glory of the Name (nāma-pratāpaḥ) is manifest in Kali Yuga (Kali-yuge prakaṭaḥ asti). One sacred merit of Kali Yuga is this: Mental virtues (mānasikāḥ puṇyāḥ) bear fruit, but mental sins (mānasikāḥ pāpāḥ) do not.

• Ramcharitmanas Uttara-kāṇḍa 102b.3-4

When such thoughts occur,

• If you do not entertain such thoughts and whenever you realise your mind has drifted you chant bhagvana naama(Saying Ram Ram, Shiv Shiv or any you have devotion towards to divert your mind from intrusive thoughts) and apologise, then no sin has happened.

• If you knowingly let the thought play itself out then you have sinned and should do prayischitta like, doing n numbers of mala of bhagvana naama, surya darshan and surya puja, vrats etc.

I personally do name chanting because in the battle field of Kaliyuga we are like Sri Lakshmana injured and on the verge of death and bhagvan's divine name is Sanjeevni Booti bringing us life.

Mahadev📿