The answer to Q1 is that there is little of anything else being produced in the Sun.
Deuterium is produced as part of the p-p chain, but is destroyed again. Similarly, there is a little bit of lithium, beryllium and boron produced in the so-called p-p II and p-p III chains, but the temperatures are so high that these nuclei are also destroyed as soon as they are created.
The CNO cycle is also a minor contributor (2%) to the energy production in the Sun. This is the catalysed conversion of protons into He, but in the process, existing carbon nuclei in the Sun are converted into nitrogen and a small amount of oxygen. The overall number of heavy nuclei stays the same but the C/N ratio decreases over the Sun's main sequence lifetime.
I'm not aware of any other significant heavy element production, though there are perhaps some cosmic ray spallation reactions producing lithium and beryllium in the outer atmosphere of the Sun.
The other questions are duplicated elsewhere (see links). In summary:
Q2: Coulomb repulsion prevents the fusion of nuclei containing large numbers of protons unless the temperatures become commensurately higher. This is feasible up to the "iron-peak" elements. Beyond that, whilst a little bit of energy could be extracted from fusion, the problem is that the required temperatures are so high that photodisintegration of the nuclei takes place.
Q3: Heavy elements are not only produced in supernovae. The main requirement is a strong source of free neutrons so that neutron-capture reactions can take place. These can be found in the interiors of giant stars produced in reactions involving carbon or neon; they can be found in the cores of collapsing massive stars as the explode as supernovae; they can also be found in neutron star mergers. The first of these is the so-called "slow neutron capture" (s-process). About half the elements heavier than iron are produced like this. The neutron capture takes place with existing iron-peak nuclei to produce heavier elements. The latter two result in "rapid neutron capture" (r-process) and it is unclear which dominates for which kinds of nuclei. The r-process produces most of the very heavy elements and all those heavier than lead.