We know that all particles can eventually be converted into energy. We know also that electrons were and still are formed by pair production from a 1MeV photon.
Do physicists have yet any idea how quarks were formed? When did that happen, before BB? Can they be produced at any time in the present? In what circumstances? Can/are they converted into energy?
Formation of quarks is well understood and they can be created or destroyed also today, like any other particle. Quarks are routinely created on particle accelerators, like the Large Hadron Collider.
According to Quantum Field Theory, particles are disturbances in quantum fields. There is a field associated with every particle (electron field carries electrons, quark field carries quarks, Higgs field carries Higgs bosons, etc). So to create any particle (like electron or quark) you just need to disturb the respective field with enough energy. In colliders this happens because you collide particles (disturbances in fields), these collisions disturb other quantum fields and this leads to creation of new disturbances (particles). In case of quarks, this leads to pair production, but due to confinement the quarks hardonize and we actually detect jets.
The incoming particles for quark pair production can be of various types. In LHC we collide protons, but quarks can be created for example by annihilating electrons and positrons. This is an example of annihilation where the products are not photons, but heavier particles. You just need to collide the 2 leptons with enough energy to allow quark production.
We don't know what happened before the Big Bang. But after the Big Bang, all quantum fields (including quark fields) were massively disturbed, so a lot of particles of all sorts were created. The underlying principle of particle creation is the same for Big Bang and for colliders: disturbing a quantum field. The difference is just where the disturbance is coming from. The problem is that particles are usually created in pairs (particle and its antiparticle). This would lead to complete annihilation at later stage. It is not yet completely clear what caused that there is now matter around us (which implies more particles produced than anti-particles at the Big Bang). This is related to baryogenesis, as mentioned by Bert Barrois in his comment.
Quarks can be converted into other particles, if they annihilate. There are multiple ways how quarks can annihilate, so a complete summary would be extensive. One example is annihilation of quarks in mesons. Mesons are compound particles made from quark and anti-quark. Certain mesons contain quark and anti-quark of the same type (like neutral $\pi$ meson), so the annihilation does occur and the lifetime of such particles is short. In other mesons, the quark and anti-quark are of different type, but they can still interact, even though it is a question whether it can be called "annihilation". Slightly more unusual interaction would be for example bottom + anti-bottom quark annihilating into Higgs boson.
Do physicists have yet any idea how quarks were formed?
There are competing theories and no definitive answers for when most quarks in the universe formed. The net number of quarks less antiquarks divided by three is called baryon number and that quantity is a conserved quantity in the Standard Model except for what are known as sphaleron processes at very high (first few moments after the Big Bang) temperatures of 10 TeV.
Sphaleron processes, moreover, aren't a sufficient mechanism to get the universe from a zero quark, pure energy state at time=0 which is what many cosmologists like to assume the early universe was like out of aesthetic considerations. But, if those are the initial conditions, then there must be some kind of baryon number violating high energy process that is new physics out there to create more quarks than anti-quarks in vast numbers, because we can't otherwise explain the baryon asymmetry of the universe (i.e. the very lopsided excess of matter atoms over antimatter atoms).
All processes other than the sphaleron process create new quarks and new anti-quarks in equal amounts.
When did that happen, before BB?
Well, we know it took place mostly before Big Bang Nucleosynthesis, because quarks had mostly turned into protons and neutrons by then. And, in a time=0 at the Big Bang cosmology, that doesn't leave a lot of time to get the job done.
Bounce cosmologies don't assume that the current Big Bang started at a pure energy state, so it may have had non-zero baryon number which is more consistent with observation.
Can they be produced at any time in the present? In what circumstances?
Particle colliders create quark-antiquark pairs on a daily basis and annihilate them, in large numbers that are very well understood mathematically. Virtual quark-antiquark pairs are created and destroyed constantly almost everywhere.
But, these interactions conserve baryon number, so the net number of quarks less antiquarks in the universe remains exactly constant.
Can/are they converted into energy?
Annihilation of quarks which are identical in all respects except their quark-antiquark character can give rise to high energy photons or gluons converting them into energy. But, in practice this rarely happens so cleanly because quarks are "confined" in quark-gluon composite particles called hadrons rather than drifting about freely by themselves.
