Dear Vipra,
I am going to try to piece together little bits of information about our tradition and philosophy and try and answer your question. If it makes sense to you please consider it.
I am a little confused about how many times the puja lamp can be lit in the home. In my home for example, if the mother-in-law lights the lamp, but by the time I come for puja, the lamp has gone out, can I light it again in the morning? Or should we wait for evening time?
These are my views only. Any form of agni is a symbol of purification in our tradition. Agni gains a divine meaning when produced with specific materials, during specific times of the day, accompanied by specific words. Why? All of the above three are different forms of energy. When they are infused together to light the lamp, the flame becomes a representation of concentration. Concentration is very important in our tradition as it is important to keep the mind focused on the kutastha chakra (third eye) to be able to have a vision of God. Even if we don't achieve 100% concentration every attempt we make towards bringing to mind to focus will bring us great benefits. The lamp/the flame is a great symbol of that idea.
So there is definitely meaning behind the idea that the lamp should be lit during Ushath kalam (right around the time the sun rises) and sandhya kalam (right around the time the sun sets) as cosmic vibrations are most concentrated in the atmosphere (thus beneficial for spiritual advancement).
So my response to your question is - if the time of the day is still part of early morning or early evening (according to whichever part of the world you live in) I would go ahead and re-light the lamp.
Also my in-laws family have the practice of putting off the lighted lamp with a flower once the puja is over, they feel it signifies that the lamp does not glow unattended. Is this ok?
In my family we have the practice of placing a drop of milk on either side of each 'mooku' or 'kaambu' before pulling back the thread gently into the oil by way of putting it off.
To the best of my knowledge such practices are meant to instill a sense of respect and care in the minds of those attending to the lamp. I don't see them as directly benefitting anybody's spiritual growth.
But if you find your specific family tradition sweet and not too cumbersome why not keep it?
I assure you there is nothing to fear if you don't put it off with a flower (or drops of milk as in my case).
But it does make one feel like the lamp is a living being that needs to be respected like God.
An appreciation of creation (including all of its materials) is just another way of praying to God. And believe me, each one of us needs every bit of prayer we can offer! One just can't pray enough!
Could someone clarify if they know the religious background of these questions?
I hope these ideas help.
Regards,
Chintana