1: I think my element is only 4500 watts at 240V. Is the problem with running the element at around 1125 watts that it will take a long time to heat my water? Are we talking like 2 hours for 5 gallons?
It will take a while, IMO you'd be better off with 240V. If you connect to your standard home outlet it's 110 or 120 V. Appliances such as electric dryers, electric stoves/ranges, require 220 or 240v power which have the larger specialized outlets (larger with 3 or 4 round prongs). So if you connect your 4,500 watt element to 120, you'll get 1,125 watts of heating power (25% of what it is capable of) and will be pulling 10.3 amps. If you connect to 240v, you'll get the full 4,500 watts and will be pulling 18.75 amps. Be sure that the amount of electricity being pulled (amps) isn't much more than 80% of what the circuit is rated for. For instance, a 5,500 watt element on 240v pulls 23 amps, which should be connected to a 30 Amp GFCI protected circuit.
2: I have done many searches on the subject. I cant figure out if I would need to incorporate a solid state relay and/or a heat sink into this system. I think I understand that a solid state relay is just a switch with no mechanical movement, therefore reducing heat. But Ive also read about people SSRs driving their heating elements and dont know what this means. Is it somehow providing more amperage for the element, and if does why is this desired?
A PID acts as a temperature controller that receives input temperature readings from a temp sensor. It sends a 12v DC signal to an SSR (Solid State Relay) which controls power to the element. You can set the PID to a specific temperature and the PID will tell the SSR to give the element full power (100%) until the temp is reached, at which point the PID simply maintains that temp. You can also use the PID in manual mode where you set the power of the element as a percentage (50% for example). This is used mostly to maintain a rolling boil (at 50%) versus a Mt. St. Helens eruption of boiling wort flying out of the kettle (at 100%). The PID sends 12v DC current to the SSR to turn the element on for 50% of the time...so in a one second interval the element would be on 100% for 1/2 second and completely off for 1/2 second and would cycle continuously like this. One thing to keep in mind is that the SSR connects the 240v power to the element...the PID simply tells the SSR when/how to apply that power. The on/off cycling of the SSR generates heat (SSR gets warm) and the heat sink is connected to the SSR to disperse the heat (consider them cooling fins) preferable outside of an enclosure.
3: I have seen my options for solid state relays (25A, 40A) but dont know how to calculate how much amperage my system would be using.
A 4,500 watt element connected to 240v will pull 18.75 amps, a 25A would be fine. A 5,500 watt element connected to 240v will pull 23 amps, a 25A might work but a 40A would be a much better choice. Keep in mind the 80% rule (actual draw of the circuit should be 80% of the circuit rating [wires, breaker]).
4: My Ranco can use multiple temperature probes connected in series and it will read the average temp of all of them. I have not seen a PID that states this as a feature. Does any one know if any/all PIDs are capable of this?
No...I'm pretty sure a Ranco can only control 120v components. You don't really need multiple temp probes or average temps, one would be fine. Besides, do you really want to be calibrating multiple probes? IMHO, get a PID with SSR & heat sink and use the Ranco on a fridge to control fermentation temps.