The text of the constitution only requires that the House chuse a speaker. It places no limit on who the house can pick. It only requires that Senators satisfy certain age and citizenship requirements, and doesn't prevent the House Speaker from being a Senator.
However there are common law restrictions on the holding of dual offices. See Dual Office Holding--Federal, State and Municipal, for example. Where the holding of two offices would cause an incompatibility, such appointments are forbidden.
The constitution does not specify the role of the Speaker, though it appears to be modelled on the Speaker of the House of Commons, the position soon diverged. The Speaker in the UK is an apolitical role, and the Speaker is expected to show strict impartiality, but by the early part of the 19th century, the Speaker in the USA was already becoming a political and party political position.
In practice, the Speaker chairs debates in the House, and the Speaker is chosen from among the Representatives. This is an example of how the constitution of the USA is more than the Constitution of the USA. There are constitutional practices and rules that are not present in the text of the constitution.
The law on Presidential Succession would continue to apply even if a Senator were appointed to House Speaker. The Speaker of the House would become President if both the incumbent President and the Vice President were removed. In doing so, they would resign from their current role(s).