I have a 2016 Synapse with a Claris groupset which now I am upgrading to 105. I have been able to find most components except crankset and FD. I might be able to survive without replacing the FD for now but my question is would the 105 RD and chain, etc. work with Claris crankset or should I just wait till I find all the components?
An 11-speed chain will provide terrible shifting on an 8-speed crankset with an 8-speed FD. The chain is significantly narrower on the outside. The usual rule of thumb is that a one-speed difference often works reasonably. A 3-speed difference is huge.
Trying the 11-speed rear with an 8-speed chain will likely be even worse, the chain will be too wide.
Try and see. Basically all bicycle chain is 1/2" pitch between the rollers, and the "speeds" difference is in thickness. You absolutely need to use the narrower chain to suit the rear setup.
As long as the claris front derailleur can overshift a little then you're probably fine. You can also squash the front mech's cage together a little and therefore closer to the sides of the chain.
I've got a MTB that was 3x5, since gone to 9 speed on the rear with the original chainrings and front mech, which works perfectly with a 9 speed chain.
The main difference is my front shifter is friction, so positioning is all down to the rider. If you intend to use the 105 front shifter with its indexed positions, then that is the likliest source of issues.
You may need to use the Claris left-hand brifter with a 105 right-hand brifter, which will exclude using a hydraulic brake on the left hand. If your left hand is your rear brake that's probably okay, if the left is your front brake, I'd think hard about priorities.
I think you could give it a try. Check to see how the 11-speed chain engages the Claris chainrings. Also check to see if the 11-speed chain will slip between the two chainrings. If it engages well and will not slip between the chainrings, you might be able to use it in the interim until you upgrade the FD/crankset. If it fails either of these tests, wait until you have a new FD/crankset to complete the drivetrain group.
If you mean to say you're putting on an 11-speed STI, no, it will not work with your 8-speed FD. Not even a little bit. Shimano road 11 front derailleur cable pull is too different from previous generations. The shifter moves too much less cable.
Pairing an 8-speed FD with an 11-speed chain also more or less will not work at all. It will be too gappy to work well.
Generally speaking, doubles are not all that sensitive to the nominal speed generation of the crank being violated. It will probably work reasonably well to keep the current crank and put an 11-speed FD and chain on it, along with the 11-speed STI. The second generation aka toggle Shimano road 11 FDs (FD-R7000, R8000, and R9100) are better about that then the first generation "long armed" style ones, which struggle to have enough total movement even when everything is right.