Solid state relay and semiconductor relay are both names of relay like device which works like a normal relay. Those are usually called also with short name SSR. An SSR is a semiconductor device that can be used in place of a mechanical relay to switch electricity to a load in many applications. Solid-state relays are purely electronic, normally composed of a low current control side (equivalent to the coil on an electromechanical relay) and a high-current load side (equivalent to the contact on a conventional relay). SSRs typically also feature electrical isolation to several thousand volts between the control and load sides. Because of this isolation, the load side of the relay is actually powered by the switched line; both line voltage and a load (not to mention a control signal) must be present for the relay to operate.
Semiconductor relays are quite common component nowadays but they are still quite expensive compared to their complexity. Typical semiconductor relay is basically quite simple circuit and you can see in this document. Modern components make it possible to build semiconductor relays using only few components and building them from components might sound a good idea sometimes (and can become cheaper). But remember that semiconductor relays which are used for controlling mains voltages must be constructed so that it is safe to use (commercial semiconductor relays are typically constructed safely).
Small DC semiconductor relays are typically used in new modems to replace bulky line relays. Those small relays are typically housed in small 6-pi DIP package. They have typical LED input like every other optoisolator and the output is typically optocoupled FET circuit. Typical optocoupled FET driving circuit works so that the LED shines to many series connected photocells which make the control voltage to FET gate to make it conductive. Typically this kind of relays have 200-300V output ratings, can handle 100-200 mA of DC current and have isolation of 2 kV or 4 kV.
AC semiconductor relays typically are constructed using TRIAC output stage and optocoupled triac driver. The triac output stages usually work at voltage range of 24-250V and can typically handled 1-4 A in small relays. There are larger relays available for higher currents (those typically need external heatsink). The TRIAC output stage in semiconductor relays typically has about 1-1.5V voltage drop when it conducts and this causes some heat generated inside the semiconductor relay (typically around 1.2W/A). Because the TRIAC output stage and some filtering components semiconductor relays have some leakage current when they are off (can be up to few mA).
Semiconductor relays typically work at 3-30V control voltage range and the input current is typically around 8-16 mA (the current does not typically change much over that voltage range).
The isolation voltage in mains controlling semiconductor relays are typically 2.5 kV or 4.5 kV. If there is bare metal for fitting heatsink, that is typically isolated from the relay electronics (check the datasheets to be sure about your relay type).
Zero-crossing turn-on and turn-off refer to the point on the AC wave form when the voltage is zero. It is at this point that an AC SSR will turn on or off. When the AC circuit voltage is at zero, no current is flowing. This makes it much easier and safer for the semiconductor device in the relay to be turned on or off. It also generates much less electrical EMI/RFI noise.
No. Because of the zero crossing circuit described above, the relay will most likely never turn on, and even if it is on, it will likely not be able to turned off, as DC voltage typically never drops to zero.
No. If the DC semiconductor relay is polarized, it may break down and conduct for the portion of the waveform that is reversed in polarity. There are available also non-polarised semiconductor relays which can be used on DC and AC but those are more expensive.
This is not recommended at all, for several reasons. First, the voltage drop across the relay will cause signal loss. Second, the conduction characteristics of the SSR are very non-linear at low operating voltages and currents. Use a mechanical relay; it will work much better.
No. There is no way to guarantee that two or more relays will turn on simultaneously when operated in parallel. Each relay requires a minimum voltage across the output terminals to function; because of the optical isolation feature, the contact part of the SSR is actually powered by the line it switches. One relay turning on before the other will cause the second relay to lose its turn-on voltage, and it won't ever turn on, or at least not until the first relay fails from carrying too much current.
TRIAC is a SCR which can operate in both current directions: from anode to cathode and cathode to anode. An SCR is a four layer diode. It has three terminals, the anode, cathode and gate. It is non-conducting from anode to cathode until a pulse of about 5-50mA is applied to the gate (referenced to the cathode). When the gate is pulsed, it turns on and conducts from anode to cathode until the current flowing through it drops below a certain level (usually about 20mA). Once turned on by a gate pulse, it cannot be turned off until current stops flowing through it.
Small ones come in TO-92 packages and can handle 500mA-1.5A. Most SCR's come in TO-220 packages and handle 4-8A, but larger ones in other packages are not uncommon. Very large SCR's can handle hundreds or even thousands of amps. Almost all SCR's and TRIACs will work at 100V and most are either 400V, 600V, 1000V or 1200-1500V.
DC semiconductor relays typically have a FET output stage which needs a control voltage to conduct. That control voltage is usually generated using the following method: the LED in control circuit shines to series of semiconductors which generate voltage from light (like solar cells). This generated voltage is sufficient to turn on the output FET.