I have a miniature Peepul tree a home. Some of them say that it is inauspicious to have peepul trees at home. I have kept it near the Tulsi plant. Please let me know if I can keep it.

We can very well grow this tree at home. It is capable of giving wisdom.

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.