Hello there,
I see nobody replied to all of your questions, and so I hope my answer will be more useful to you than the others replies you have received.
1.)The warmest places in Portugal are the Algarve and the Alentejo region (southern and eastern parts of Portugal).
2.)The area best for relying on English is definitely Algarve, as it is full of Brit °µÍø½ûÇøs.
3.) Sites to find cheap rental for long term: try the site which combines all the searches from a number of real-estate agencies, and then just pick your region, adjust the pricing information and search.
Then if you find something you like, you will see the name of the agency involved in the real-estate at the bottom of the casa sapo page, so you can contact them.
For 400 euros it will be a bit tough to find a T3, but it is doable in the suburban areas and not very friendly like neighbourhoods. House is out of the question for the price, only an apartment can be managed in Algarve. I think the best option for you is to come and see first how you like the region. Algarve is next to the sea, it is full of tourists and °µÍø½ûÇøs. It is possible to find cheaper apartments there, because a lot of buildings were built very fast due to the arrival of many °µÍø½ûÇøs, so the buildings are very badly done and with low quality, but, in return, the pricing of such apartments is good. Now, the other region-Alentejo is soo hot, that people have moved out of it and it is not very inhabited. In Alentejo there might be a possibility of finding an old T3 house for your budget, but you won't have anything nearby, esp.not schools. That region is very dry and hot, with maximum temperatures in Portugal during the summer. It is also close to the border of Spain. A notable city of this region is Evora, it has universities, so it would be good for kids (but no T3 for 400 euros can be found). But, as far as English is considered, you won't find many locals speaking it in rural areas of Alentejo. Somehow, your budget and the likeliness of people speaking English are confronted when combined together. More English speaking residents means that area is more touristic, and therefore, the pricing is higher than the average.
But, if I may ask, why would you like to move the kids from Spain, if they already speak some Spanish? Have in mind that the international schools are cheaper in Spain (with English language) than they are in Portugal.
Good luck!