Yes, a baby's Haji is valid and the parent earns reward — use a carrier instead of a stroller, plan for cooler hours, and bring all supplies.
Yes, performing Haji with a baby is boleh. A woman brought her child to Nabi (shallallahu alaihi wa sallam) and asked: 'Is there Haji for this one?' He replied: 'Yes, and you will have the reward' (Muslim). The Haji performed on behalf of a child is valid and earns reward for the parent, though the child will still need to perform their own wajib Haji after reaching puberty.
Practical considerations are significant: the extreme panas and massive crowds during Haji present real challenges with an infant. Use a front-facing baby carrier rather than a stroller for tawaf and sa'i, as strollers are nearly impossible to navigate in peak crowds. Bring ample supplies — diapers, formula/food, medications, sun protection, and a lightweight blanket. Plan your rituals during the early morning or late evening when temperatures are lower and crowds are thinner. Keep the baby hydrated frequently. Diaper changes and breastfeeding breaks may require stepping away from the ritual temporarily — this is completely boleh. Ensure your Haji group is aware you have an infant so they can provide appropriate support.
Source: Sahih Muslim, Kitab al-Haji; Ibn Qudamah, al-Mughni