Sorry you're going through all this.
It's surprising that you have been advised to introduce food early. Babies usually have a growth spurt at about 4 months, and start to feed more frequently as a way of building their mum's milk supply (breast milk is made on a supply and demand basis). More frequent feeding is only very rarely a sign that they are not getting enough milk - it's just that they are 'placing their order' for the next few weeks of growing. If your baby was 6 weeks premature, then maybe he is having his growth spurt now.
Your health visitor probably sees babies who are having a growth spurt on every day of her working life, so she is used to dealing with this issue and I can understand why she is bemused. Is she concerned about your baby's weight? It's normal for babies to drop a centile or so, so long as they are still gaining weight overall.
Waking twice in the night is totally normal for a baby of five months old. Their tummies are still tiny and it's much more unusual for a baby to be sleeping through at this age! Don't worry, they all sleep through eventually 
If you are happy to eliminate problem food from your own diet, there's no reason why you can't continue breastfeeding for as long as you want to. It's not easy following a restricted diet, but it is possible. I think wheat is a particularly tricky one, but I did manage to exclude dairy, eggs, nuts, peas, lentils, beans, sesame, mustard, bananas, onions, garlic, tomatoes, citrus and a few other things from my diet for a long time. My view was that if it was good enough for ds then it was good enough for me, and that it was good practice for cooking meals that he could eat.
Eczema can be caused by many things, but my ds's eczema was totally related to diet and cleared up when I realised what he was allergic to. Might be worth keeping a food diary.
In answer to your questions:
- I wish I knew! I found these boards as helpful as anything else!
- They are all the same, so pick any brand of formula you like. If any ingredient is proved to have a beneficial effect, then it has to be included in all brands by law. If there is any ingredient in one brand that is not in the others, then there is no real evidence that it has a beneficial effect.
- It sounds like you know what you are doing (e.g. introducing low risk foods first, and very slowly). Talk to your hv about this.
By the way, breast milk is much more high calorie than most weaning foods (ten times more calories than baby rice or pureed carrot) and is also a complete, balanced food. The same is true of breast milk substitutes.
Good luck. .