In the event of martial law being sanctioned in the United States, I am in question about the Constitution and how it would apply. If a soldier comes to my home during martial law and asks to be quartered or be let in for an "unlawful" search am I still subject to outline the Constitution to soldiers and deny them from quartering in my home or searching in my home without a warrant signed by a judge? I know the legislative and judicial branches are in the backseat during martial law, so that's why I ask this. If the 3rd and 4th amendments didn't apply then could they waltz right into my home even if I said they cannot?
Martial Law is defined:
"the imposition of direct military control of normally civilian functions of government, especially in response to a temporary emergency such as invasion or major disaster, or in occupied territory." Source: Wikipedia.
Martial law /noun
the law temporarily imposed upon an area by state or national military forces when civil authority has broken down or during wartime military operations.
the law imposed upon a defeated country or occupied territory by the military forces of the occupying power.
The primarily important aspect of Martial law is that it is imposed by the military, most often with an occupational force and not through a civil process.
In the U.S. where martial law may be imposed by the executive branch of the government, there are statutory limitations in how this can be done lawfully; including Congressional authorization, and limits on what can be imposed. I would refer to the US Constitution's protections of Habeas corpus, the "Posse Comitatus Act", the "Insurrection Act", and the Bill of Rights amendments you cited.
If done outside of these limitation, this would be extralegal and would be more similar to a coup d'état by one branch of the government to seize power over the entirety of the Union or a military overthrow of the government by force. Therefore, the Military's governance is not restricted by any preceding legal or political doctrine. In the United States, this would include a nullification of any part or the entirety of the US Constitution and Bill of Rights, based on the disposition of the Military command.
Or it would be illegal, resulting from a fracturing of the Union - a civil war. In this situation, any legal precedent is potentially null and void in the regions controlled by those who have splintered from the current government. Within those areas, the rules practiced would be set at the leader's own discretion. However, in the case of an illegal breakage from the government, soveriegnty is not acknowledged and is arguably never established. Therefore, the original law of the land would still be in effect as long as the original government was maintained, and violations could be prosecuted as such when original control is reestablished.
The Third Amendment says:
No Soldier shall, in time of peace be quartered in any house, without the consent of the Owner, nor in time of war, but in a manner to be prescribed by law.
If we are at war, then a law could be passed to allow forced quartering of soldiers. I don't think a martial law would suffice; there's really no point to saying "in a manner to be prescribed by law" if the military can just make up the law as it goes along. But if we're already to the point of declaring martial law, I doubt anyone would stop them.
The Fourth Amendment states:
The right of the people to be secure in their persons, houses, papers, and effects, against unreasonable searches and seizures, shall not be violated, and no Warrants shall issue, but upon probable cause, supported by Oath or affirmation, and particularly describing the place to be searched, and the persons or things to be seized.
Whether a search is "unreasonable" is a matter of some interpretation. There are already cases where the police can enter your home without a warrant and it's not considered unreasonable - for example, if they have reason to believe you're actively destroying evidence. So, whether the military can search your house might depend on why they're doing it. If it's "we detected radiation from an atomic bomb somewhere on your block, and we need to find out where it is before it explodes" then they can legally search your house without a warrant. If it's "we're searching all houses in this area just in case we find something", then they legally can't, although then you still have the problem of how you tell the guys with guns that they can't do something.
Under martial law, the laws governing the military now govern the civilians, at least that's the general principle. Each nation, or more correctly, each military, can decide how to apply that.
If it were to happen in the US, that means the civilian population would fall under the jurisdiction of the Uniform Code of Military Justice, which does not implement all aspects of the US Constitution. When one joins the military, they agree to give up some of their constitutional rights, for the duration of the enlistment.
Presumably, the 3rd and 4th amendments would no longer apply, although the commander on site would be well advised not to exercise the full extent of the UCMJ, if they wanted to avoid an armed insurrection.