There is a verse in Quran addressing the spider's home as the weakest:
The example of those who take allies other than Allah is like that of the spider who takes a home. And indeed, the weakest of homes is the home of the spider, if they only knew. 29:41
Two questions:
As a an intro I just found this commentary maybe it helps:
All the above-mentioned nations were involved in shirk, and their belief about their deities was that they were their supporters and helpers and guardians, and had the power to make or mar their destinies; so when they will have won their goodwill by worship and. presentation of offerings they will succor them in need and protect them against calamities and afflictions. But, as shown by the historical events cited above, all their beliefs and superstitions proved to be baseless when their destruction was decreed by Allah. Then no god, and no god incarnate, and no saint or spirit, and no jinn or angel, whom they worshiped came to their rescue, and they met their destruction with the realization that their expectations and beliefs had been false and futile. After relating these events, Allah is now warning the mushriks to the effect: "The reality of the toy-house of expectations that you have built on your faith in the powerless servants and imaginary deities, apart from the real Master and Ruler of the universe, is no more than the cobweb of a spider. Just as a cobweb cannot stand the slightest interference by a finger, so will the toy-house of your expectations collapse in its first clash with the scheme of Allah. It is nothing but ignorance that you are involved in the web of superstition. Had you any knowledge of the Reality you would not have built your system of life on baseless props. The fact is that none other than the One Lord of the worlds in this universe is the Owner of power and authority, and His support is the only support which is reliable. "Now whoever rejects taghut and believes in Allah has taken a firm support that never gives way. And Allah hears everything and knows everything." (Al-Baqarah: 256).
This is the tafsir point of view.
Scientific research showed lately that it is the female spider who builds the home not the male, and in the Verse of the Quran the female conjugation of the verb to take/to make (a dwelling/a home) in Arabic ( اتَّخَذَتْ).
After that the female spider search for a male to fertilse her then she kills and eats the male.
Therefore some scholars say this is the reason why this home is weak. Its weakness isn't a material weakness because the material of the web is one of the strongest in nature. A synthetic copy of it is known as Kevlar. But the weakness mentioned here is the lack of peace, mercy and love which makes a good and strong home according to the Verse 21 in Surat Ar-Rum (Surah 30).
If you read the Ayah carefully, it says:
The example of those who take allies other than Allah is like that of the spider who takes a home. And indeed, the weakest of homes is the home of the spider, if they only knew. 29:41
I highlighted the parts that is relevant to your question.
"those who take allies other than Allah"
"spider who takes a home"
Home means a place of safety, a spider's alliance is with its web, its home, and this is related to those who ally themselves with those other than Allah (swt).
Ibn Kathir explains it much better than me (may allah be pleased with him):
“This is an example given by Allah, the Almighty, of the idolaters who revere gods besides Allah. The idolaters hope that those gods will assist them, provide for them, and they turn to them in times of hardship. In this regard, they are like the spider’s house, in its weakness and frailty, because by clinging to these gods they are like a person who holds on to a spider’s web and does not gain any benefit from that. Had they known this, they would not have taken protectors other than Allah. This is unlike the Muslim believer, whose heart is devoted to Allah and, in addition, does well in following Allah’s decrees. The Muslim has grasped the most trustworthy handle, the one that never breaks because of its strength and stability.”
The above should answer your first question.
As for your second question, it is very clear that the spider's web is very easily damaged, at the same time, does not provide the spider with protection from the environment. Compare a spider's home, to any other living creature's home for reference.
Your question looks interesting and important. I said it owing to the word "take" that has an extensive meaning in English. So, occasionally this is difficult to have a precise translation from Arabic into English, but totally may be one of the most suitable meaning of take which has come in its translation, is "Choose" or even "getting"
As it has come in the Surah 29: 41:
مَثَلُ الَّذينَ اتَّخَذُوا مِنْ دُونِ اللَّهِ أَوْلِياءَ كَمَثَلِ الْعَنْكَبُوتِ اتَّخَذَتْ بَيْتاً وَ إِنَّ أَوْهَنَ الْبُيُوتِ لَبَيْتُ الْعَنْكَبُوتِ لَوْ كانُوا يَعْلَمُونَ (عنکبوت/41)
The example of those who take allies other than Allah is like that of the spider who takes a home. And indeed, the weakest of homes is the home of the spider, if they only knew.
Your other question (In what terms is the spider's home the weakest? : Perhaps, the meaning of the mentioned verse about to the weakness of the spider's home (which is made by web) is up to the firmness and strength of it which is so weak; therefore this won't be counted as a safety place for the person who is trusting it at all.
I personally think that the verse كَمَثَلِ الْعَنْكَبُوتِ اتَّخَذَتْ بَيْتاً وَ إِنَّ أَوْهَنَ الْبُيُوتِ لَبَيْتُ الْعَنْكَبُوتِ لَوْ كانُوا يَعْلَمُونَ apply to both land and water spiders. This verse find a literal translation in the case of water spiders which "take " water bubbles as houses as it is scientifically indicated that water spiders live their whole lives in water bubbles (see this link:https://www.youtube.com/watch?v=jjFew5Lk2r4). This verse also applies to land spiders though in a different way. The land spiders here build their houses which though built from strong threads, the house itself is still weak. The verse describes the house as weak and not the thread and the weakness of the house lies in the fact that the house doesn't protect it from the environment around.For example, the house doesn't protect it from rain and it is 100% transparent and this makes it a prey for its enemies such as birds. To conclude, the status of the spider using its weak house as a shelter is the same as the status of a person asking a weak human(no matter how strong human is) for support and help. As indicated above the weakness of the spider's house can be understood (100%) literally in the case of a water spider ( a house made of a water bubble), and this weakness can also be understood (100%) literally as well in relation to land spiders in the case when a spider faces environment difficulties and enemies where its house doesn't help it and it leaves its house and run to holes for shelter.
