Question - What is the backing for the statement that she killed Kolasura or any other asura for that matter?
The reference to Mahālakṣmī killing Kolāsura is found in the Skanda Purāṇa Verse 4.1.5.(73-78).
Chapter 5, Pūrvārdha, Kāśī-khaṇḍa
ततो व्रजन्ददर्शाग्रे पुण्यराशिस्तपोधनः । चंचच्चंद्रगताभासां भाग्यवानिव
सुश्रियम् ॥ ७३ ॥ विजित्यभानु नाभानुं दिवापि समुदित्वराम् ।
निर्वापयंतीमिव तां स्वचेतस्तापसंततिम् ॥ ७४ ॥ तत्रागस्त्यो महालक्ष्मीं
ददृशे सुचिरं स्थिताम् ॥ ७५ ॥
- to 75. Proceeding ahead from there, Agastya, the mass of merit, the storehouse of penance, saw in front of him Goddess Mahālakṣmī (of
Kolhapur) like a lucky man, the best of glory. She had the lustre of a
hundred rising moons. With her own lustre, she appeared to surpass
even the Sun, even during the daytime. She was brilliantly dazzling,
(yet) she appeared to alleviate the distress of his (Agastya’s) mind.
रात्रावब्जेषु संकोचो दर्शेष्वब्जः क्वचिद्व्रजेत् । क्षीरोदे
मंदरत्रासात्तदत्राध्युषितामिव ॥ ७६ ॥
- Lotuses shrink and close during nights. During New-Moon days the Moon goes away somewhere. There is the fear from the Mandara mountain
in the Milk Ocean. Hence, it appears as though she has put up her
permanent residence here (Kolhapur).[8]
यदारभ्य दधारैनां माधवो मानतः किल । तदारभ्य स्थितां नूनं
सपत्नीर्ष्यावशादिव ॥ ७७ ॥
- Ever since Mādhava retained her with great honour, she had certainly stayed here as though out of spiteful rivalry with her
co-wife.
त्रैलोक्यं कोलरूपेण त्रासयंतं महासुरम् । विनिहत्य स्थितां तत्र रम्ये
कोलापुरे पुरे ॥ ७८ ॥
- A great Asura was frightening the three worlds in the form of a boar. The goddess killed that demon and stayed in the beautiful city
of Kolhapur ever since.
English Translation by G.V. Tagare
Further, it is asked -
Ques - Who is the composer of this ashtaka? Is it Indra as its starts as Indra uvācha or someone else? Where else does this get referred to?
Although most online or offline resources refers the śrī mahālakṣmyaṣṭakam to be a part of the Padma Purāṇa; a case similar to the Śiva Gītā, which is said to be from the Uttara-khaṇḍa of Padma-purāṇa, however, currently no extant recension of the Padma Purana contains Śiva Gītā.
Similarly, I couldn't find śrī mahālakṣmyaṣṭakam in the currently available version of the Padma Purāṇa.
So most probably, śrī mahālakṣmyaṣṭakam stava was a part of some non-extant version of the Padma Purāṇa, just like the Śiva Gītā.
And
The speaker and author of the aṣṭakam is Indra-deva, which is quite clear from the starting and concluding lines of the composition:
'Indra uvāca (इन्द्र उवाच)'
Meaning = Indra said:'
&
'itīndrakṛtaṃ śrīmahālakṣmīstavam (इतीन्द्रकृतं श्रीमहालक्ष्मीस्तवम्)'`
Meaning = thus ends the Indra authored śrī mahālakṣmyaṣṭakam.
