Both my boys were ebf, and breastfed well into toddlerhood. One clearly had allergies by the time he was a few weeks old, and the other seems fine so far. The one with allergies was given formula in hospital when he was born.
There was a study (in Australia, I think) which appeared to show that breastfeeding did not protect against allergies - but those babies were mostly given formula in the maternity ward so that their mums could sleep, so that is hardly conclusive. Can't find the link right now, or link on my phone, but can dig out out if you're interested.
The nhs have recently removed "helps prevent allergies" from the list of benefits of breastfeeding on their website.
Early introduction of solids is another issue. .. and again, the evidence is inconclusive. I did read about a study which showed that introducing high risk foods such as peanuts early actually helped to prevent allergies developing (not that this is any help of your child is already reacting to traces in your milk!)
Both my boys had their first tastes of food at 5 and a half months when they grabbed food from my plate. Some babies are ready earlier than others, and the six month 'rule' is only a guideline. If they can sit up unaided, have the gag reflux developed and are helpingthemselves to any food they can reach, then they are ready.
I'd love to see some real evidence either way. My instinct is that breastfeeding is proven to be good for the immune system generally, so it must surely help, but maybe it's more complicated than that. ..