Recommendations for a month in Scotland

Well you could pretty much choose any town in Scotland, so much depends on your interests. Although if you want to be in a location served by tour operators such as Rabbies, then Edinburgh, Glasgow, Aberdeen and Inverness are their main hubs.

Condos do not exist in the same way in the UK as in the US. We typically just refer to properties as houses or flats. It might be hard to find a reasonably priced month long rental as most 'short term lets' in the summer months are done a week at a time, and you are unlikely to get any sort of reduced rate for taking a month as there is no shortage of customers looking for one or two week rentals.

Whenever we have wanted a longer term stay, and especially the 'live like a local' aspect we have sought a home exchange. We used Homeexchange.com to arrange our swaps and have been all over Europe, the US and Canada. It worked well for us as we had the knowledge that someone would be in our home and it wouldn't be empty while we were away. We did plenty of video calls and emails with the prospective exchange partners before committing and in over 30 exchanges we never had a bad experience. We also swapped cars where practical. Might be an idea for you to consider? Plenty of listings in Scotland.

Scotland's geography is challenging in many areas. Transportation time (whether driving or using trains/buses) may be considerably longer than you anticipate, and public-transportation frequencies may also present challenges. I think you'll find spending the entire month in a single location rather limiting. Aside from the lovely rural areas, both Edinburgh and Glasgow have a lot of local sights. For me, it would get very old, making repeated day trips to those cities.

A month in one place, in July, as Skyegirl says, might be hard because that is peak tourist season.

Also being tourist season, if you are in one of the main tourist cities from whence tour buses run then it is a bit of a strange amalgam as you aren't in some ways living like a local if you are taking tour buses to where all the tourists go. This is an interesting amalgam of ideas.

To my mind you want to be somewhere out of the tourist hotspots, apart from anything else that is being more respectful of the locals (it isn't the phrase I would have used, but I can't quite put my finger on the right phrase).

If I was a local then I wouldn't be using the likes of Rabbies.

Say I was a resident of Dumfries and Galloway in the South West of Scotland I would be using local companies for my trips out such as Houstons Travel (as an example). If I was a bit further north in say Ayrshire or the northern part of the Borders, towards Glasgow and Edinburgh then the likes of Strathmore Travel- they are a Scottish company and do lots of very interesting day trips to worthwhile places off the beat of all the well known companies. And straight up- I use Strathmore for multi day trips. If I lived just 40 miles further north then many of their day trips are very enticing, but I am just out of area, and that extra hour at start and end of day makes all the difference.
If I (personally) had more money I might also be looking at companies like Solway Tours- and I know they do quite a lot of bespoke private tours for Americans as I see it in their social media. Again that is just an example of who is out there.

A month is a long time in one place- even if it gives you a chance to become involved in local clubs, volunteering and churches, known in the cafes etc. If your Mum actually was a local then she would get a free bus pass for the whole of Scotland (for all I know you might as everyone over 60 gets it)- but not for 'temporary locals'.

People who know me know what I am going to say next before I say it- two weeks somewhere in the Scottish Borders and two weeks somewhere in Dumfries and Galloway (possibly not Dumfries- maybe somewhere a bit more central like Kirkcudbright) could work well. There are good bus services especially in D and G, however they have just been re-tendered and a total dogs dinner was made of that process. The result is that the previous superb value daily, weekly and monthly county rover bus tickets (the monthly especially at about £4 a day was huge value) have gone. So using the bus is now expensive- but for locals who have to use the bus they have to live with it, and the huge extra cost on household budgets.
For a temporary local bite the bullet and rent a car for the duration. The Scottish Borders are in a much happier place regarding bus tickets.
Dumfries and Galloway is big enough that you could even split it- two weeks in the East and two weeks in the West- in Stranraer or it's hinterland.
Both areas have huge opportunities for day trips, walking and historic sites- most of which are just not in the international (or even national) spotlight.

I spend a lot of time in Scotland and mostly agree with what has already been said.

Instead of one place, think of splitting it up into two or three places to make day trips easier. (The country is bigger than you think.)

Look into day tours. Even the locals take them. I was recently on a 3 day Rabbies tour and 5 of the 16 passengers were from Scotland.

Rabbies, Timberwolf and Discover Scotland are three local--as in Scottish--tour companies that have good reputations. Look into where they go, decide what you want to see, and consider whether you want to leave the driving to them or do it on your own. Be aware, in some areas, the roads are narrow and even one lane.