What if travel destination is Egypt? Can they see if a warrant has been issued from the USA?
In general, if someone is traveling while a country wants to arrest him, the big concern is an Interpol notice. These are formal requests from one member country to all others with regard to individuals of interest.
In particular, the Red Notice is issued when a country is willing to extradite someone from anywhere else in the world. This is very expensive, so countries don't always do it. Interpol notices are checked at immigration controls and usually during police checks in a country as well.
If you have had a Red Notice issued, you should expect that you will be detained at the port of entry of almost any country you visit, likely for several weeks, possibly for months, while the country which issued the notice works on extradition proceedings.
Some Interpol Red Notices are public, and there is a searchable database available for these.
It's important to note that a Red Notice is not an "international arrest warrant". There is no such thing. It also does not compel member countries to detain or to do anything with respect to the noticed individual. However, many countries will detain and notify the country which requested the notice to begin extradition proceedings, depending on what treaties they have in place. In other cases, the noticed individual may merely be refused entry. And it's possible the Red Notice could be entirely ignored (but don't count on it).
If you've attracted this level of attention from the US, it's likely that your passport will also have been revoked, and a notice to that effect made available to the rest of the world. This also will make travel difficult.
If a country doesn't want to spend all that money on extradition, they can also issue a Green Notice, which advises member countries of an individual's criminal activity, but doesn't request extradition. Immigration officials will also see a flag as soon as they read your passport, and will become aware of your criminal history when they look into the flag. You may then be refused entry to a country based on your criminal history or based on deceiving the immigration officer about it (if you attempt to do that). Many countries do not allow someone with outstanding arrest warrants to enter.
There are also country-specific agreements which give certain countries direct access to criminal records of other countries.
For instance, the US and Canada share access to law enforcement information, and if you cross the US-Canada border, Canadian border officials can access your complete US criminal record. This includes anything a US police officer can see, not just Interpol notices. In this case there may not be any extradition; you'll be refused entry and returned immediately to the US, where you'll then be arrested.
If you have a warrant issued in the US, it is only valid in the US jurisdiction; unless the US issues an international arrest warrant for you, through Interpol.
If it does that though, rest assured your passport will also be flagged and most probably revoked and you would immediately be detained and proceeding will start for deportation.
So, if you simply have a US arrest warrant, this information is not available to Egyptian officials - since they only have access to their own databases and the Interpol database of notices and wanted people.
Even if they can "see" the warrant, they cannot detain you under it. They can deny you entry because of it - some countries do not allow people with any criminal history to cross their borders; but then you would be held for illegal entry or possibly fraud (for hiding your criminal history from immigration), and not for the offense on the warrant.