When exactly do we use different mudras?
In Hinduism, Mudras are extensively used during worship rituals. Mudra is an inseparable limb of the worship ritual.
The first step in a Shodashopachara Puja is Dhyanam, which itself is done using a Mudra called Kurmamudra.
The next step of the Puja is Avahana (invocation) which again is done with the help of 6 Mudras namely -- Avahani Mudra, Sthapani Mudra, Sannidhapanimudra, Sannirodhanimudra, Sammukhakaranimudra and Sakalikaranamudra.
A screenshot from a book called "Nitya Karma Puja Paddhati" shows few of these Mudras:

All other subsequent steps of the ritual are also accompanied with the use of various Mudras like -- Ankusha (for invoking presence of Holy rivers in the pot), Avaghunthana, Matsyamudra (for sealing holy pots), Dhenumudra (for Amritakarana or for making ingredients Amrita like) etc. ... Finally everything ends with Visarjana, where again a Mudra called the Samhara Mudra is used .
In Gayatri worship, the use of 24 Mudras (Sumukham, Samputam, Vitatam etc.) is widely prevalent. The screenshots (from the book "Nitya Karma Puja Prakash"), given below, show few of those Mudras.


In this way, we have Mudra representations of everything like we have Mudras for deities (Shiva Mudra, Durga Mudra, Ganesha Mudra etc; given in Brihat Tantrasarah and other texts).
In Nitya Shodashika Arnava Tantram one entire chapter is dedicated to Mudras (the title of the chapter too is "Mudra"). In this chapter, Lord Shiva provides the descriptions of how to form the 9 Mudras related to Shri Vidya worship.
The Panchavayu Mudras viz: Prana Mudra, Apana Mudra, Vyana Mudra etc. are also shown during Naivedya ritual (i.e. while offering edibles to the deity). The scriptural verses containing the depictions of these Mudras and how to use them in a ritual are given in this answer.
But all these information are not much useful for the kind of questions you have asked.
So, I am using the book "Yog Sadhna Evam Yog Chikitsa Rahasya " by Swami Ramdev as a reference.
This book has a separate section on Mudras. It gives the health benefits that are associated with the these Mudras.
Few screenshots from the book are given below:


The questions that you've asked are answered in this book as follows:
Mudras start effecting the body immediately.
If a Mudra is formed with the left hand then that immediately starts effecting the right side of the body and vice versa.
Mudras can be done while being seated or while standing. Even while walking one can do them (with the exception of Shunya Mudra).
But if done while being seated in regular Yogic postures like Padmasana, Sukhasana etc., the Mudras yield more benefit.
One can start doing 10 minutes sessions of Mudras and then can increase the duration to 30 or 45 minutes. One can also have multiple
sessions if covering the required duration is difficult to achieve in a single
session.
Only things to remember here is that Surya Mudra can be avoided when the weather is hot and Shunya Mudra mustn't be done while walking.