Is wiping the penis with a dry tissue after urinating sufficient for purification (tahara), and does the dry penis touching the underwear after wiping necessitate purifying the garment, or is wiping sufficient without washing?

You must be careful of waswasah (whispering doubts from Satan) and extremism in purification. Lingering in the restroom for a long time is harmful, both health-wise and religiously. The best way to perform istinja' (cleaning oneself after relieving oneself) is to remove the impurity with tissues or similar materials, then wash the area with water. Water alone is better if one wishes to limit oneself to one method. Wiping with tissues alone is sufficient as long as the urine has not exceeded the usual area. However, if it has exceeded that area, then only water is sufficient. Touching the penis after it has dried to a dry garment does not defile the garment, and it is not necessary to change clothes or wash the feet after urination unless you notice impurity when intending to pray. Urine should not be withheld due to its health harms and its violation of the prohibition of praying while suppressing the two impurities (urine and feces).