"1. How I can determine whether they are hot water or steam? "

Steam radiators will have a small silver 'bullet' shaped valve on one end (opposite the valve and pipe entry) to allow air out and steam to enter in one pipe steam systems.

A small knob on the bottom of the vent allows some adjustment of the closing time of the valve and how long steam is allowed to flow before the valve closes off (based on temperature) and stops the flow of steam through the radiator.

There are a few two pipe steam systems in residential use, but they are rare ad often use commercial style convectors (with an electric fan in many cases) instead of simple radiators.

When the system is operating you can check for cracks.

Was the house allowed to freeze with water in the lines?

A one-pipe steam system typically does not have much water in the lines, they are designed to drain the water back to the boiler (though with age and settling there may be water pockets that d\form and interfere with correct operation.
The boiler is what needs to be checked for freeze damage since they DO contain standing water.

A hydronic (water) system would be filled with water and subject to serious damage from freezing (even steel pipes split, let alone radiators).
A compressor and adapters to use air to pressurize the system would be in order.

There are not usly any good ways to repair radiators damaged by freezinf.
The parts for any particlur design are very hard to find in most places, and welding cast orin cracks is itsown special nightmare.
The thin cast iron of radiators often creates other cracks from expansion and cooling when you try to weld.

There are few shops set up to heat an entire radiator hot enough for welding, and radiators often use lead seals between segments that would melt out long before the iron was hot enough for safe welding.

Either new radiators or search around for old ones in good condition.

Don't forget--if you're new to having radiators--they need to be bled each season before you start using them in the fall. That involves turning a little valve (we need to use a screw driver) to let the old, yucky water drain out. We just put a paper cup under the cock to catch it--it's maybe a half cup for each radiator.

Enjoy! The heat is SO much better than any other, IMO. With radiators, we're comfortable with the house between 55 and 60--in the old house with the drafty hot air system, we had to keep it between 68 and 70 to feel comfy in winter

"if you're new to having radiators--they need to be bled each season before you start using them in the fall."

Not required for steam, they bled themselves every time they are used in the typical one pipe steam system.

A well sealed hydronic system should also not require bleeding.

They are sealed systems.
The only way for air to get in is if water gets out and is replaced with new water with dissolved air.

The only time I needed to bleed any of my multiple hydronic systems was after working on them and having to add a lot of fresh water with its dissolved air.

If you find that you have a steam system:

Here is a link that might be useful: We Got Steam Heat! - A Homeowner's Guide to Peaceful Coexistence

Pipes pic. Sorry they're all sideways. I took these with my phone.

Valve pic.

Valve pic

Looks like hot water type. The small thing at the top which you designated a valve is the bleeder valve; you open it to let the air out of each rad so water can fill it.
Casey

Hot water based on the typical bleeder in the picture.

It takes a small key with a square recess to open the valve to let air out.

An actual hardware store should have the key.

buy more than one or you will be forever looking for the thing.

Every radiator has to be bled when the system is first filed with water.

You need something to catch a small amount of water.
The water will often have a definite odor from circulating pump oil and be black.

Start with the lowest one and wait for water to come out.

Close the valve and go to the next highest radiator.

If the system is not leaking you should only have to do this once every few years.

You WANT the old water to stay there as long as possble.

This helps to limit corrosion in the system (once the oxygen dissolved in the water is 'used' in corrosion the corrosion stops).