Crores of animals are killed during some feasts. Why are the poor innocent animals forcibly killed everyday for no mistake of them. Why the animals are suffering in the hands of humans? When will this stop?

Ahimsa is the foremost Dharma There is no doubt about it.
You are talking about animals suffering in the hands of humans. But we barely realize that all of us are suffering in the hands of Karma (fate). In order to get answers to the question of ‘what for’ and ‘why’, you should try and understand God. That is the only way. Until then, whatever I say would only be a consolation.

Some people kill mosquitoes. Is it not a sin? Can you list all the sins that the human does and atonements for them?

Practice Ahimsa as far as practically possible and chant Mahamantra all the time. Instead of analyzing the swept dirt, why not just find a way to destroy them?

Basically, isn’t killing a living being a sin? But we use repellants to ward off, in fact kill the mosquitoes, and traps and rat poisons to kill rats, and we hit cockroaches when we see them inside our homes. Now is this right? If these aren’t done they prove to be a big menace for us.

All the main religions (viz. Hinduism, Islam, Christianity) propagate non-violence, however, allow to some extent sacrifices in the Name of God. The speciality about Hinduism is that, it is permitted and prohibited according to the way of one’s life. A householder is allowed to kill beings in order to protect himself and lead his life, whereas when a renunciate (‘Sanyasi’) kills even a small living being even if it afflicts pain on him, it is considered a grave sin.
In general, according to our Shastras, it is sin to kill harmless and helpless living beings. They prescribe that by doing so, one would be afflicted with skin diseases in his next birth. It is advisible to do the minimum amount of harm to other beings to lead a peaceful and happy life.

Why some people offer animals to God as sacrifice? Does God really like’s those offerings?

Even with respect to Gods, there are three types of Gods – Sattvic, Rajasic and Tamasic. Rajasic and tamasic Gods are offered animal sacrifices and alcohol. Sattvic offerings are done to Sattvic Gods.
In general one inherits the qualities of the God whom he propitiates. So it is always advisable to propitiate Sattvic deities.

What kind of impact would it have on my karma if I kill an insect, or a fly? Is that a sin?

The dharma prescribed varies as per the Ashrama. that is, if one is a Sanyasi – (a renounced person), the prescribed norms are very stringent. If he kills an insect even if it harms him it is a sin. He is not allowed to do so. but for a householder, it is different. If an insect harms him, he can kill it, if necessary. That would not amount to a sin. but if he does it purposefully or for pleasure it is a sin.
Shastras state that if a person kills helpless beings like insects and flies, he will contract skin diseases in his future birth.
In short, a householder can kill an insect as an act of self-defense if harmed by it. It is not a sin.

We always say that killing living beings is a sin. But we use silk by killing silkworm and silk is used as part of many rituals. What is the significance then?

It is true that killing living beings is a sin and the Vedas rightly say so. However, in some cases, if the benefit that the person and the society derive out of such a killing is significant, it shall be excused.
During a child’s Upanayanam (sacred thread ceremony), at the time of Brahmopadesam (when the ward is initiated into the sacred Gayatri Mantra) by the father or the Guru, they are covered in a silk cloth. Likewise, silk is also used in other ceremonies and rituals. The reason behind this being, silk has a unique property of retaining the radiations of the Mantras.
Giving a little more thought into it, even consuming certain plants and vegetables (like spinach or greens, tubers etc.) amount to killing lives.
Hence, it is generally accepted that in instances where such a harm is inevitable for survival, do the minimum harm possible.

We are studying Hinduism and recently learned about reincarnation. My friend said, “If Hindus believe in reincarnation then why don’t they kill animals for food? Doesn’t it help them to achieve a better birth each time until liberation because they are sacrificing themselves to feed you?”

Extending your friend’s logic, if killing animals is better than killing plants, the best way is to better kill humans themselves! It sounds very illogical isn’t it? Tell your dear friend that death is not a guarantee for a better birth in our religion!

Red and black ants have infested my pooja room and consume the offerings meant for the Lord as well as the ‘kolam’. Is it okay to use insecticides to keep their numbers from swelling?

It is certainly a sin to kill ants in the pooja room. Scriptures say that if one kills small helpless creatures, one is bound to get skin diseases in the next birth.
The very act of drawing ‘kolams’ is so that ants and cows can feed on it. So it is not advisable to use insecticides in your pooja room.