How old is he? It is very unlikely to be his heart unless he is at least 2 or 3 and even then, that would be extremely unusual to be showing symptoms.
When he makes the breathing sound is it kind of like wheezing, then after a bit it stops? This may just be 'the cavalier snort', which is normal and often happens to some cavaliers. It is a result of their nose being extra short. All of mine do it now and then; some dogs do it a lot. Next time he does this, try gently tipping his head down towards his chest while you cover his nostrils with a finger. If it is the snort he'll likely stop right away and carry on without a bother. I know of people who have rushed dogs to the ER in the middle of the night having heard the snort and thinking the dog couldn't breathe.
They can often do it after or even wghile eating -- Leo somteims does -- probably because of excitement and the position down and forward of their heads.
Sounds like he might have some allergies or could have picked up kennel cough and hence the sneezing. Dogs scratch for all sorts of reasons from ear mites to fleas to (in cavaliers) PSOM. When the scratching is particularly aggressive or odd it could be syringomyelia but that doesn;t sound like an SM scratch to me.
Rescues often have ear mites etc.
Regarding food allergies: don;t let the vets try to sell you Hills or another expensive diet to try. Just feed something like James Wellbeloved and get one of the non-poultry flavours (dogs tend to have more allergic reactions to chicken and beef, and to grains, than other things).
I am guessing the sneezing (and maybe the coughing) is just a mild cold or a bit of kennel cough and these are not allergies, and that he's just doing the cav snort. But do have him checked; I'd just be cautious to rush into treatments for what may only be some basic cavalier behaviour.
Pm me if you want to let me know where he came from and his background. If he was only recently into rescue then they often are a bit under the weather for a bit due to stress and the changes -- esp. if he was in a pound or rescue kennel.