May one use a disposable cup for kiddush (literally, "sanctification," is a blessing recited over wine or grape juice to sanctify the Shabbos and Jewish holidays), or must it be a silver cup?

The material for a cup used for kiddush, can be of any material, e.g. gold, silver, copper, glass etc. It is however ideal to use a nice, respectable cup.

Regarding disposable cups, the Rabbis have long debated if it considered a utensil and thus acceptable, or since it is meant to be disposed of, it isn't acceptable.
 
Rabbi Moshe Feinstein ruled stringently, and did not allow it except in cases of great necessity where no other cup is available. This is also the ruling of Rabbi Yitzchok Weiss. He goes so far as to necessitate "designating" this disposable cup as the one you will always use in the future when no regular cup is available.
 
Other Rabbis, including the Tzitz Eliezer and Rabbi Shlomo Zalman Auerbach are more lenient and consider a disposable cup an acceptable utensil. However, he rules that although it's acceptable, it is lacking in "Hidur Mitzvah - the beautification of a mitzvah" to use such a cup. 
 
If using the hard plastic cups available nowadays, which are much more durable than the regular disposable cups and indeed are used by many at respectable meals, many Rabbis feel that these are indeed acceptable.
 
Some people, when using a disposable cup, put one cup into another one and make kiddush using this double cup. It isn't clear exactly how or why this would solve the problem.
 
For a final ruling a Rabbi must be consulted.

Many people use ear plugs for various reasons during the day or when they sleep to block out noise.

Regarding their use on Shabbos, it would depend on what kind of earplugs they are.

An earplug that is like a small sponge which does not assume the shape of the inner ear when inserted and reverts back to its original shape upon being removed from the ear, may be used on Shabbos.

However, earplugs that are made from silicone, wax or similar materials which assume the shape of the inner ear that they are put in to and retain that shape when being removed may not be used on Shabbos

If one is driving on a journey and is delayed for example in traffic, can one daven (pray) in the car whilst driving?

If one must stop to daven and it is pouring with rain outside or it is a bad location and one is worried for their safety, can they daven inside the car or must they exit to daven?

Davening (praying) whilst driving a car is absolutely prohibited, as it is impossible to concentrate on the road and on davening at the same time. 
  
If you pull over to the side of the road it is preferably you should go outside and stand at least for the Shemona Esrei part of davening.  If that isn't possible either due to rain or if it is a dangerous neighbourhood, then you may indeed daven sitting down in your car.

All of the above is provided that you are running against the clock and will not be able to daven normally when you get to your destination safely.

A father must educate his children to do all Mitzvos (Torah and Rabbinical) that are appropriate to their age.

One also needs to prevent a child from doing any sin which he is capable of relating to, starting with telling the truth, eating only Kosher and not carrying things on Shabbat (in areas where one may not carry, i.e. where there is no Eruv.)

A child who stole must return the theft if it exists.  If it no longer exists then he needn't return its value even after he becomes an adult. However, in heaven he will be required to account for the theft, so it's recommended to make amends.