Yes, but that's not a good idea to edit [Basic Paragraph]. Even David Blatner says to leave [Basic Paragraph] be.
Simply create a new Paragraph Style as your preferred default.
In case you don't know, the way to do that is to open InDesign from it's applicaiton icon, and from that empty interface (in other words, don't create a new page, and don't open InDesign from a recent document), Opt/Alt click the new paragraph style button and create you new default style and give it a name. From thereonin, every time you open InDesign, your new style will be there in the palette below [Basic Paragraph]
The only problem with editing [Basic Paragraph] is that the text will change to match a different definition of [Basic Paragraph] when you move it to a new file with a a different definition. This in no differnt thatn waht would would happen with a style named Body Text, or any other name. The only reason it pops up is that by default every documten since CS3 (or maybe even earlier, I've long forgotten) has a stle named [Basic Paragraph], and unless the user adds another style that's waht all text will use.
I don't recall what Mr. Blatner said about [Basic Paragraph] and it's actually irrelevant as long as you know how styles work, and how "based on" styles can be affected. Most users I know will tell you never to base another style on [Basic Paragraph] because it can have unintended effects when moving between files, precisely because users are free to use any definition of [Basic Paragraph] that they choose.
A new default style is fine, but it certainly isn't required.
It looks like Paragraph Style is where the problems are coming from. However, opening ID on its own and making a new paragraph style did not solve the issue, because as soon as I open a document, the style I made disappears from the list. So I made the new style while already within an existing document, and that makes ID no longer revert to Minion Pro. Unfortunately, it looks like that just makes it choose something different to revert to. If I choose anything different from the font/size/etc. specified in the current paragraph style - say, if I want Tahoma 12 for a little bit instead of the usual TNR 11 - it'll add a "+" next to the style in the paragraph style window, and tell me that the style is currently being overridden by whatever changes I chose, and I can alt-click to clear it. I paste something in, press Enter... and the "+" disappears, with the style overrides (i.e., the font/size/etc. changes I specifically chose a few seconds ago) gone.
Is there a way to tell InDesign to just keep whatever font/size/etc. I've chosen at any given moment, even after I paste text in and then put in a carriage return?
That's just the way Indesign works. I think you need to get more familiar with the way ID functions. Many people here recommend Sandee Cohen's book Visual Quickstart Guide as a good start. Peter already gave you an answer regarding changing the default font to what you want. Any new document will start with just one paragraph style: Basic Paragraph. You are suppose to create (or import) the styles you need in your document.
Let's see if we can explain this better...
Here's what I think is happening based on what I've been able to do here to reproduce this.
First, in the Clipboard Handling prefs the setting is to paste text only, not All Information. The TNR setting is a local override to Basic Paragraph which has been set as the default in the character panel or control panel, but [BP] itself is still set to Minion Pro.
You can type all day in ID with settings like this and not have the font change to Minion Pro, so it is tied to the paste settings. I think what is happening is that ID must ignore any formatting in the copied text in order to make the pasted text match what is set at the cursor position and for some reason resets the default text attributes to the paragraph style definition, so when you press enter your local override default is gone, but only for that frame. I believe this is actually using the Paste Without Formatting command.
It doesn't seem to matter if you continue to type in the same paragraph, the next one loses the override. You can make this happen using the Paste Without Formatting command after copying text you type with your default settings as an override, as well, even if the format of the copied text is already exactly the same because you typed it in the same paragraph. The trigger for this though, is the position in your text where you paste -- it only happens when you paste at the end of the existing text in your story.
I'm not sure if this is a bug or intended behavior, though I'm leaning towards the latter as it makes some logical sense to me. Paragraph formatting for the next paragraph is picked up from the last character when you press enter as you type at the end of a story, whereas if you type on the end of an exisiting paragraph in the middle someplace, the break at the end of that paragraph is already formatted.
Bottom line, I think the only way to make your workflow work is to make a default paragraph style with your TNR formatting eihter by editing [BP] or making a new one. I'm not sure why you didn't have success with a new style. Did you create the new style with nothing open? Did you set the formatting to be TNR? Did you also be sure to highlight this style with nothing open so that it was set to be the default?
What I said above is is accurate for the condition described -- clipboard handling set to text only, but there are other scenarios that might be worth exploring. First would be the clipboard handling setting. If the text is already formatted in Word the way you want it, changing the setting to All Information would result in the next paragraph picking up that format from the Word text rather than reverting to [Basic Paragraph] without overrrides.
But the bigger question in my mind is if the copy/paste workflow is actually the best method for what you are doing, and why you don't use File > Place... Are you grabbing random chunks of text from multiple files, lots of chunks from one file, all of the text? There are lots of options available for preserving or discarding formatting or mapping to new styles in the Import Options when you place instead of paste.
I do a newsletter for one client where the articles are authored by various different members of the organization whose skill level in Word ranges from the pathetic to the not-so-good. Lots of the articles have been styled in odd ways to make them "look good" or make some sort of organizational sense to the author, but they have no resemblance to the styles actually used in the newsletter. I usually place these as-is in ID and fix the formatting there by applying my styles or running scripts to remove stuff that doesn't belong. I find that much faster than anything else I've tried, even if I'm going to copy/paste from the fixed text in a frame on the pasteboard to get it into position in the document layout.