How can I use 2 styles like this:
style1: 1. Initial Trust Principal
style2: As the initial principal of this…, etc
With no line break between them; they just flow one right after the other.
There is a similar question here, but without an answer: [Writer] Style that automatically bolds first sentence of a paragraph?
What is missing in the question is the intent with the bold underlined text. Obviously, the paragraph is numbered, as a list or a heading?
With a list, just use a specific character style.
In case of a heading ending up in a TOC, the workaround is more complex. Search this site for “inline styles”, like this question containing a link to a tutorial.
I remember having seen a solution based on a standard heading, but hidden, inserted as a cross-reference at head of the “body” paragraph.
Thanks! I thought there might be a standard solution. I first used this in Ventura Publisher on CPM so I thought by now someone would have it working. 
This is not possible in LO, possibly because of the ODF standard itself (I don’t remember). There are some workarounds depending on what exactly you are trying to do.
For example, you may be able to use two character styles for the heading and details, or perhaps just a character style for the heading and none for the rest (so that you can still use emphasis, etc. as usual). Obviously, you still need a paragraph style for the paragraph margins and spacing.
Thanks for the feedback.