Your legal position is very poor here but all the same you must seek proper legal advice asap with regards to the children, finances and the property.
Basically what is his is his and what is yours is yours. If the property is solely in his name he is indeed within his rights to ask you to leave.
If a house is bought in joint names (either as beneficial joint tenants, or as tenants-in-common) then it should be split accordingly on separation, and either party can force a sale of the property to realise their share.
If the parties are contributing unequally to the purchase price, or to payments on the property, then this should be reflected by being designated as tenants-in-common and holding unequal shareholdings (say 70% and 30%), rather than the equal shareholdings of beneficial joint tenants.
If the property is in the sole name of one party then basically it remains that person's property on separation, unless the other party can establish that there was a common intention that they would be entitled to a share in the property. How do they do this? Here are a few examples
It may have been agreed in a simple conversation (proving it tends to be the problem!), or in writing between the parties at some time
If the other party has directly contributed to the purchase price the courts are likely to accept that at least part of the property should have been in their name
If there has been an "understanding" between the parties and the non-owner has acted to their detriment as a result (eg contributed to mortgage repayments, paid household bills, or, perhaps, sold their own property) then the courts may agree they should share in the property.
The parties can, of course, come to an agreed settlement, but if not, such disputes can become messy and expensive.
My guess too is that there is another woman involved.