a child will not let themselves starve. this is what i told myself over and over again when my son was the same.
i saw DR. OZ on OPRAH and he mentioned that MANY kids are "white" kids... white bread, white rice, white spaghetti.... that was my son to a T. it sounds like your son too.
i can tell you that with perseverance, it will get better.
1) cut out biscuits & cake. you don't want him to fill up on garbage so that he's too full for real food.
2) don't fight him. i know this is hard and you want to keep trying new things. put out something for him to eat. if he doesn't eat it. take it away. do NOT offer anything else. it sounds cruel, but he will learn very fast that he needs to eat.
3) try to make food as bland as possible. THAT is why they are "white kids". children have over 10,000 taste buds. adults only have 3,000. so a lot of kids shy away from strong tastes and only stick to the very bland food.
he likes white rice, then boil some chicken and shred it up. not small chunks, but small shreds. it will be more difficult to see that there is something besides rice. maybe fry some rice with egg whites, white chicken breast and a little pureed cauliflower. that way, he's getting everything he needs AND it's all white. hard to see that there's anything besides rice in it.
4) hide the fruit and veg. if he likes cake... make him carrot cake! if he likes chocolate cake, add cauliflower puree (it sounds gross, but you REALLY cannot taste it!), make banana bread, add apple sauce to other sweet dishes.... then he's still getting his fruit and veg.
5) dipping... does he like to dip food? if he likes chicken nuggets (my kids adore them!) when you cut the chicken into pieces, dip in egg, then carrot or cauliflower puree, then bread them and fry/bake them. you can't even tell that the veg is there. try making your own dipping sauces like ketchup...that way, you can control the sugar, but he still gets to dip!
6) cheesy pasta! my kids LOVE it! we sometimes use the kraft mac & cheese, but add our own things, like pureed broccoli or cauliflower. we also add extra "real" cheese.
.................
as for getting sick more easily.... that's normal too. is he going to playgroups? the playground? the indoor playroom in the clubhouse? if so, he's going to get sick more often.
if the doctor says not to worry... believe him.