Seeing from his date of birth and the year in which the Battle of Karbala took place (source: Wikipedia) he was 3-4 years old at the time.
At Battle of Karbala he was a small child and was with Caravan of Imam Hussain a.s. and he was inside the tent with the women and other children. Women especially Sayyedah Zeinab a.s. (the sister of Imam Hussain a.s.) were caring from children. All of the men with Imam Hussain a.s. who were supporting him were killed by army of Yazid and finally Imam Hussin a.s. himself was killed. then the army of Yazid fired the tents and women and children escaped from the tents into the desert with firing cloths and then the soldiers arrested the women and children as captives and took their belongings like jewelries and then took them with the head of Imam and other killed companions of Imam to the Castle of Yazid at Damascus, Syria.
The presence of Imām Bāqir (as) is also something that historians agree upon since there are many reports that state he was born before 61AH.
"Imām Bāqir (as) was born in 57AH in Madīnah."
Similarly, when a person asked Imām Bāqir (as), "Were you there in the era of Imām Ḥusayn (as)?" the Imām replied, "Yes".
Imām Bāqir (as) says in respect to his presence at Karbalā, "My Grandfather Imām Ḥusayn (as) was killed when I was four years old, and I still remember his being killed."
In accordance to this selection of accounts, there is no objection (or obstacle in proving) that Imām Bāqir (as) was present in Karbalā, and naturally since his eminence was quite young at the time he lived and travelled with his family. Also since he was very young at the time, the books of history have mentioned very little information about his presence in Karbalā.
About Ali Asghar(as)(6 months old) dropped himself out of his cradle I could not find this mentioned in authentic historical Maqtal books. You are requested to mention the reference of it to can be investigated.
About Imam Sajjad a.s. it is recorded in different Maqtal books that:
He was sick but came out of tent but could not take the sword. Umm Kolthum was calling behind him: return O son. he replied: my aunt! leave me to do Jihad with the son of messenger of Allah and help him. Imam Hussain a.s. said: O Umm Kolthum! inhibit him and take him back to tent because the earth should not be empty of prophecy of Muhammad peace be upon him.
Reference: 1. خوارزمی ، مقتل الحسین ، ج 2ص 32 (Kharazmi, Maqtal ul Hussain, vol. 2, page 32)
About Mohammad ibne Ali (as) responding to that sidaye isteghasa I could not find anything in reliable Maqtal books.
Note: Maqtal literally means the place/time of fighting/killing and is said to the historical books describing the details of a war specifically the battle of Karbala. More info about Maqtal: http://en.wikipedia.org/wiki/Maqtal_al-Husayn