Westie problems after adoption

I would like some advice please on some issues with a 21 month old Westie named Benji . We adopted him 5 weeks ago , we are the fourth owners. I will give the background story . He was born in Co Cork Ireland , date August 2015 , he has a EU pet passport which shows  all the details he was microchipped in December 2015 and also vaccinated for rabies and anti echinococcus treatment, but no stamp to show he was ever left Ireland 

The passport shows he was then sold to a new owner in the UK, who I  am assuming had him for around 12 months , then  someone the owner knew offered to buy him due to been tormented by the owners 3 children . The new owner unfortunately realized after 2 months she was unable to give him quality time , as she worked full time , and could only take him out for 10 minutes in the mornings, and ten minutes in the afternoon and evenings , so put him up for adoption on a well established dogs for sale online. My husband and I made contact with the lady and she thought we were the ideal couple for Benji, so we bought him 5 weeks ago. He was slightly underweight when we got him , but that has been remedied, as I  cook him homecooked food , minced turkey or chicken, with veg , or fish etc. He gets taken out for 2 half an hour walks every morning , then 1 hour or more in the afternoons at our local beach or park , then 2 short walks each evening. Lots of attention and love, but we are having issues with some of his behavoir.

The problems he has are the following, when  he sees our neighbour through the windows,( in the  shared  garden, ) he starts to whine and cry , and does not stop , he runs to the door and goes ballistic,we have done all the right things to stop this as advised ,even taking him outside to see her ,  he then jumps up as though he has never had any affection, which is of course not true.   Nothing is working,I am assuming it was because he was locked in the previous owners house nearly all day and when he saw her coming home he was over excited. The other concern is he does not bark or signal in any way  to go out when he wants to poo , and have had a few accidents in the last week , we take him out very regular as explained , and he does poos then , and rewarded. Apart from these 2 incidents he is a happy boy , and has settled in well. Any good sound advice would be greatly appreciated . 

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Thank you Joannne for your sound advice.