Intention is one of the conditions of the wudhu, but it is not obligatory to recite it. The intention is an act of the heart and not verbal.

It is a condition in wiping the head that the wet hand is wiped over the dry head. If it is not possible to do so because of the wetness of the head, then one should wait until it is possible.

There is no precise limit for wiping, but it is sufficient that the wetness on the hand is not mixed with the wetness in other parts of the wudhu.

If you are sure that you have passed wind during any part of the prayer, the prayer is invalidated as is the wudhu.

If the wetness on the head was such that it transfers to the hand and leads to wiping the feet with that wetness, then this invalidates the wudhu, since it is obligatory to wipe the feet with the wetness of the wudhu. If the wetness on the head was such that it does not transfer to the hands when wiping the head, then there is no problem in wiping the head in this case.
Therefore, it is better to dry the head before the wudhu.

If he has a specific period in which he can perform the wudhu and the prayer without passing wind, he should wait for it and perform the wudhu and prayer within this period.
If he does not have such a period, then he should perform the ablution before the prayer and offer the prayer, and when he passes wind during the prayer, he should suspend the prayer and perform the wudhu and continue where he stopped. Based on an obligatory precaution he should not do anything that invalidates the prayer when performing this wudhu, such as talking and the like. However, in such a situation if one does anything that conflicts with the form of the prayer or turns away from Qibla, for the purposes of performing wudhu, it will not invalidate the prayer. If one has such a condition, he should make appropriate preparations to ensure that the acts of wudhu during the prayer are minimal.

It suffices in the correctness of the wudhu to perform what is considered as wiping, even if it is on one toe.

If one doubts the correctness of the wudhu but at the same it is probable that he washed the necessary parts without intending to do so, then he can assume that the wudhu is in order and he does not need to repeat the prayer. If he knows that he did not wash a necessary part of the wudhu, then it is invalid and he must repeat the prayers that he performed with such wudhu.

If the doubt is conventionally normal, and not a type of waswasah, one should pay attention to it, and therefore wash until he is certain that during the washings the water has reached all parts of the face or arm. If the doubt is a type of waswasah, then one should ignore it.

One can recognize waswasa by some signs, like when one is having his doubts in situations much more than other people in similar situations. Also, one can determine it when his inner feeling is that the religious worship is correct, but he still has worries and tensions that prevent him from accepting his internal certainty.

You start by washing the face from the topmost part of the forehead where the hair begins to grow to the farthest end of the chin, and its breadth is what is covered between the thumb and the middle finger. The said area must be washed fully. Then, wash the arms by starting with the right arm from the elbow to the tips of the fingers. Then wash the left arm in the same way. Then, wipe the head using the wetness of the right hand. Then, wipe the feet from the tips of the toes to the ankles, starting with the right feet, then the left. It is sufficient to wipe normally and it is not mandatory to cover the mentioned wiped areas fully with wetness.

Yes, the wudhu is void if the wiping was with new water. If the water used in wiping is mixed with new water, and therefore the wiping was done using both waters, then the wudhu should be repeated based on an obligatory precaution.

Touching the skin of the opposite sex does not invalidate the wudhu.

Wudhu when wiping using the said water is not valid. One can use those fingers that have not come in contact with new water or the palm of the hand to wipe the head and feet.

If he joins between the two prayers as in the question, he does not need to repeat the wudhu. If he separated between the prayers, he should do the wudhu again before the second prayer.

It is not necessary to use the hands in washing the face and arms during wudhu. It is valid to wash them by placing them under the tap directly, such that they are washed from the top downwards. It is also a condition of the wudhu to wipe the head and the feet with the dampness on the hand that originates from the wudhu process only, and not to allow the hand to become wet from other than that before wiping.
In light of the above, before one proceeds to wiping the head and feet, one should ensure that a part of the left arm is unwashed before using the right hand to wash the left arm, in order for the right hand to have only the water of wudhu on it. Otherwise, the water on the right hand might be extra water and wiping with it will invalidate the wudhu.

Yes, it is obligatory to make up the past prayers.

The mentioned is not what the tradition means. What the tradition means is observing the religious guidelines when performing the wudhu and not to cross them, for example: wiping replaced with washing, or wiping the bottom of the foot instead of its top, and similar cases where the religious restrictions are transgressed.

If wiping with one finger covers the obligatory area of the wiping, which is from the tip of the fingers to the ankle, then it is sufficient.

Yes, it is true, and you can wipe normally and not touch the wet places.