If you touch any part of the dead body or bone of a dead body directly with out gloves, you need to perform Ghusl of touching a dead. If you touch wearing gloves, then you don't need to perform Ghusl of touching the dead.
Touching any Najis item including dogs or its urine etc does need Ghusl at all. You need to wash the part which touched the Najis item with water, then perform Wudhu and pray. It is a sinful act to delay performing the Prayer till it's time goes. Performing the Qadha namaz will not omit the sin of not praying within the time of the prayer.
You must perform your Namaz within its time under any circumstance.
Ghusl does not need water flowing on the body, but needs just making any amount of water reach to every part of the body even by wiping water on all parts of the body. A big scholar in Najaf Ashraf who is Ayatullah Sayyed Muhammad Husain Al-Hakeem said: Ghusl can be performed with one cup of water by putting my hand in the cup of water then wiping over my head, face, neck, right side of body then left side of the body. Making the water touch by wiping to the whole body won't take more than one cup of water.
You can refer to Islamic rules of your Marja' of Taqleed for more details.
Taharah is a real condition شرط واقعي in Prayers and fasting. Any worship which requires Purity (Taharah) is invalid if it was done with out purity.
If you could not learn basic rules of Ghusl and never been informed that Ghusl is obligatory in certain cases, which is very rare now a days, then the rule is different. But if you knew about Ghusl and it becomes obligatory in certain situations, and you did not know that it applies on you, and you fasted with out Ghusl, then you need to perform the obligatory Ghusl then perform every worship which was performed with out Ghusl.
You can read the verdict of Ayatullah Sustani as follows:
Ruling 1593. If a junub in the month of Ramadan forgets to perform ghusl and remembers after one day, he must keep a qaḍāʾ fast for that day; and if he remembers after a few days, he must keep a qaḍāʾ fast for all the days that he is certain to have been junub on. For example, if he does not know whether he was junub for three or four days, he must keep qaḍāʾ fasts for three days.
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