I have combined a trip to Ireland and Scotland in both 2005 and 2012 and with 7-10 days total I would highly recommend sticking to just one at a time. The trip we did in 2012 was 13 days and most of it was Scotland. Scotland was the goal of the trip and the only reason we spent anytime in Ireland at all was we determined the best way to get to Edinburgh from Boston was to go through Shannon. Since we were going through Shannon we thought it would be worth it to spend 3-4 days in Ireland and visit some of our old favorites. In hindsight we could have used that extra time in Scotland.
If it were one of my first few trips to Ireland then I would have wanted to devote the entire trip to Ireland. If you do foresee being able to make a separate trip to Scotland in the future then yes I would keep them as separate trips. Both places warrant the full attention.
That being said, it's doable and I didn't find the extra hassle overwhelming just two extra half days of hassle catching flights, dealing with airport security, luggage retrieval, etc. I've learned over the years and many trips to minimize these types of hassles as much as you possibly can.
BTW, I did write up a trip report on this site in the Trip Reports section about our visit to Ireland and Scotland in 2012. Just rereading parts of it reminded me that not having enough time in Scotland and having much longer drives than expected was tiring. To visit Scotland and do it justice you need as much time as you can possibly manage.
Welcome to the forum. I have been to Scotland as well as Ireland and must agree that with the time you have available you are better off choosing one country. Even with 7 to 10 days you will only see the highlights of either Ireland or Scotland. Plus you will lose most of a day traveling from one to the other.
I suppose it all depends what you aim to get from the visit.. There are tour bus itineraries that cover Scotland and Ireland in 10 days.. would you want to spend 9 hours a day flitting from place to place?
Rather than the broad sweep of Ireland (2 countries) and Scotland, where are the places that draw you to the plan.. knowing the key points is a start to defining the time needed.
__________________
Two rules for Ireland, Take your time & bring a sense of humour
Once you know if it will be 7 or 10 days, perhaps you will have a better idea of what you want to concentrate on. Some people like to be on the go continually. Others prefer a bit saner trip. Which you choose is up to you and your personality.
This is a trip I would like to take. On the bucket list is Dublin north to Northern Ireland to see the Giants Causeway and then go to Scotland. I would do at least 10 days. Is it feasible to hit the high lights of Northern Ireland and the high lights of Scotland in 10 days?
It has been so long since I was in Scotland I really can't comment on if the highlights could be seen in perhaps a week. Last time I was there it seemed the Highlands were vast with a lot of driving space between places. Maybe make a "hit list" and run it through an online mapping program to see if feasible.
As for No. Ireland you can probably see what you want in 4 or 5 days. Depending on the "list".