"Do you need to go to a probate company or probate solicitor to distribute an estate?"
No, if the person who died left a will then it is the executors named in the will who are responsible for sorting everything out and applying for probate if necessary.
If the person died without making a will then their nearest surviving relative (spouse, then children, then parents, then brothers/sisters, then grandparents, then aunts/uncles) can apply to administer the estate and apply for probate.
"Can money in a savings account be distributed to people named in a will before a house is sold?"
Short answer, "It depends". Long answer, "It does really depend on so many different factors that you need to speak to the executors"
But, there is nothing stopping the executors from distributing one part of the estate (eg a bank account) before all of the estate is distributed.