Could an object be stiff but not rigid or vice versa?
When is each one used?
And what is the opposite of each of them?
In engineering mechanics a rigid body will undergo no deformation whatsoever under any amount of force.
In contrast, stiffness is a measure of how much force it takes to deform a body by a given amount.
In general terms something is said to be stiff when it has a high stiffness coefficient, i.e., it takes a lot of force to deform it; but notice that it does deform.
As the stiffness goes to infinity the body becomes rigid, i.e., no amount of force will be able to deform it.
In practice, nothing is truly rigid, but many times assuming a rigid body simplifies the math needed to analyze a system and when the stiffness of the body is high enough the effects of the assumption are negligible.
Therefore in practice, I reserve the use of the word rigid for cases where there is no possibility for bending (whether figuratively, or literally) and use the word stiff when I believe bending is possible, albeit only with a large amount of force.
As for opposites:
opposite of rigid - deformable : able to be deformed
opposite of stiff - flaccid : not firm or stiff
stiff, adj., n., and adv.
Rigid; not flexible or pliant.
Of the body, limbs, joints, muscles, etc.: lacking suppleness, unable to move without pain (esp. owing to age, cold, injury, disease, exhaustion, etc.).
rigid, adj. and n.
I. Stiff, firm, unbending.
Of a person or part of the body.
The simplified and generalised answer is that, in modern usage, if it usually moves but now doesn't, it is stiff. If it doesn't usually bend, it is rigid.
To answer your questions, yes - something can be stiff but not rigid. A door-hinge for example, when you have to heave on the door to get it open, is stiff but not rigid. COCA gives one lonely hit for 'stiff hinge' and none at all for 'rigid hinge'. Interestingly (given that COCA is American English), Ngram disagrees, and for American English says rigid hinge is more used, but for British English it is a stiff hinge. This reflects how much of a grey area this is...
The reverse is true too. A pole would be described as rigid if it doesn't bend, but not usually described as stiff. COCA gives two hits for rigid pole, and one for stiff pole. Ngram agrees with this one., at least for the past 60 years.
It's unfair to the average English learner but the answer is not that simple.
Stated simply, the words are not always synonymous. Different fields of learning have defined the words differently to distinguish between different qualities.
Depending on the domain and context, appropriate meaning will apply.
Free from all context, in general English writing, the words have been widely used interchangeably to imply a resistance to changes (esp., in shape).