If for whatever reason you can't get it to work, you can get a USB HDD dock. This one looks like it can do SATA or IDE.
I'd say your best bet is to install an ethernet card or a usb card. You could also pull out the hard drive and attach it to a modern computer via an external hard drive adapter, like this: https://www.amazon.com/Inateck-External-Reader-Applicable-SA03001/dp/B08BNPXSJ9/ref=asc\_df\_B08BNPXSJ9/
Could be because windows 7, I still use Win7 myself actually and am running into issues using newer parts xD Though its hard to pin point the exact problem there, I'm no tech guru by any means, you'd likely have to take it to a professional for the blue screens. That being said, so long as it isn't the hard drive causing the problem (which it sounds like maybe it is) you could, once you have the new system, drop that drive in temporarily with something like this to help with that.
Far as part picking goes, I know a lot of those services provide a basic means of choosing compatible parts, if the one you plan on using doesn't for whatever reason pc part picker has one, it is a little more basic I think but it should do the trick for finding compatible parts. Far as what parts you may actually want in your system is definitely something you would have to look at for yourself, based on what you are telling me though I would guess you could get away with a lower end pc. Something with a Ryzen 5 or an i3 would likely suffice you for a long time.
To put it in perspective I've been running parts that are half as good, if even that, as what you will see now. And Likely still only maybe half as good as what is in your system now, the only thing really stopping me from playing newer games is windows 7, granted I can't play them at max settings and often time it looks like a potato, it still works. So unless you change your mind and start doing higher end work you won't need higher end equipment.
And I will not disagree with you, a laptop is way easier to bring in then a full tower, I've had to bring mine in a few different times and that sucks, like dragging a 500lb brick around. Even my newer build with lighter parts in it (because it isn't 15 years old) is still fairly heavy.
Like I said I'm not real familiar with laptops, so it could be likely that Asus makes non gaming type and Dell does make gaming types. I know Dell/HP and other larger brands are more business oriented and tend not to go crazy with extra stuff like the latest graphic cards. Where as Asus (honestly can't think of any others that make laptops) is more focused on the gamers/enthusiasts and tend to have a smidge extra in their builds. Far as Acer goes I remember them being a lower end build, though I could have just been buying older systems when I came across those. Good quality just much lower end then I would have personally used.
No problem! I know it can be frustrating to try and figure out what parts you need or if you should do this or that with little to no input from others. Just glad I can help in any way.
As stated, remove the HDD and connect it to a USB device so that you can read it on another PC. The manufacturer's website will have a manual you can use that explains how to remove the hard drive (some laptops back then had a compartment on the bottom with 1 screw that allowed for quick access to it)
I use an adapter like this https://www.amazon.com/Inateck-External-Reader-Applicable-SA03001/dp/B08BNPXSJ9/ref=asc_df_B08BNPXSJ9
Also, using a bootable USB stick for a Linux operating system would work but this, to me, requires so much more learning because of the way Linux and Windows differ in even some simple file access methods.
Personally, i have this one
Unless you need a dual bay + all the memory stick readers.
https://smile.amazon.com/gp/product/B08BNPXSJ9/ref=ppx_yo_dt_b_asin_title_o05_s01?ie=UTF8&psc=1
Or you could get an actual hdd enclosure.
https://smile.amazon.com/gp/product/B08BNPXSJ9/ref=ppx_yo_dt_b_asin_title_o00_s01?ie=UTF8&psc=1
you will need to pull off the adapter in the 2.5" to expose the true IDE pins.