When it is said that everything in this world is God’s creation then why people reject certain days as inauspicious? When we have faith in God and carry out our duties as prescribed and adhere to the religious path, do we need to give importance on these things like not undertaking any thing on Ashtami and Navami and Prathamai days?

God created poison too. Can one consume poison just because it is also God’s creation? The great Maharshis have prescribed that good things be done in certain times and avoided at other times.
It is only for our benefit that they have prescribed it. It is up to you to either believe it or reject it.

Krishna was born on Ashtami day and Rama on Navami day. Then why do we believe that Ashtami and Navami days are inauspicious?

Those that are applicable to normal men are not necessarily applicable to Gods.
It is because of astrological and astronomical factors that these two days are deemed as inauspicious. According to our scriptures, the days are numbered according to the state of the moon. ‘Prathamai’ is the first day after a full moon day or a new moon day (‘prathama’ means ‘one’ in Sanskrit.). ‘Dvithiyai’ this the second day, ‘Truthiyai’ is the third day and so on. Thus, ‘Ashtami’ and ‘Navami’ are the eighth and the nineth days after the full/new moon day. The number ‘8’ is associated with Saturn which is a planet with ‘tamasic’ qualities. The number ‘9’ is associated with Mars which is associated with Mars, a ferocious planet.
Hence these days are considered as inauspicious for men.
Such numerological and astrological factors are limited by time place and person. Hence they are not applicable to Gods.

Generally it is said that we must not prostrate before a Mahatma when he is in the midst of his japam/dhyanam as it would deprive him of his tapas. Does it apply to Avatara Purusha also?

According to the Shastras, we should not prostrate before someone
who is sleeping
who is lying down
who is just about to leave somewhere (such a namaskaram is done only to a corpse)
who is performing Japa
who is carrying a baby
who is having some ‘theetu’
who has just been served food (this is done only during Srardha)
There are other such instances, but these are the most important ones. This applies for everyone.

What is ‘Poorna Kumbham’? To whom should it be offered?

As the name suggests, ‘Poorna Kumbham’ signifies a full ‘kumbham’ (pot). It is a traditional way of paying respects to great Mahatmas and Kings while welcoming/ receiving them to one’s place.

What is ‘Kaaraam pasu’? What are its features?

‘Kaaraam Pasu’ is a cow that is pitch black with not even a speck of another color on its skin. Another feature of this kind of cow is that it would have a small hump on the posterior part of its neck.

Why is it that we are advised not to cut nails or hair on auspicious days?

The Shastras believe that there is enormous ‘prana shakti’ (life force) in the nails and hair. Hence we are advised not to cut them during auspicious days.

Some people say that one should not take bath or wash the legs after returning from a temple. Why so?

As per our Sanatana Dharma, we generally wash after we visit a place of death or inauspicious happening. When we visit a temple, we carry with us the positive vibes of the charged cosmos there. So it is advised not to wash it away.

For human beings, auspicious events are linked with the start and inauspicious events are linked with the Thithi. Why are all auspicious events related to God linked with the Thithi (eg. Vinayaka-Chaturthi, Rama-Navami, Karthik-Sashti, Krishna-Ashtami, etc.)

“Thithi” is nothing but date. Instead of a thirty day month, in olden days, it was divided into two fortnights – the bright and the dark. The fortnight starts with ‘Prathamai’ (‘pratham’ means first), ‘Dvithiyai’ (‘Dvithiya’ means second), ‘ThrithIyai (‘Thrithiya’ means third) and so on and culminates in Amavasya or Poornima (new moon or the full moon respectively), each of which denote the stages of the moon.
So Thithi is nothing but the date, and the star is determined based on the day.
For gods, some give importance to the star (like Punarvasu for Rama or Rohini for Krishna) and some attach importance to the Thithi (like Navami for Rama and Ashtami for Krishna).
There is no specific reasoning behind following one way or the other.

When can we give alms to the beggars outside a temple – before entering the temple or while returning?

It is good if you donate on your way back.

I have holy lemons and coconut. May I drop them in a river like Cauvery?

Yes you may do so.