Using elements to build slideshow on 1 pc and view on another

In the past, if you purchased a version of Elements within 30 days of the release of a newer version, you would be able to upgrade to that newer version for free (after jumping through some hoops).  I don't know whether that policy is still in effect.  If you were to purchase Elements 2020 now, I don't know if you would be eligible for a free upgrade.  If I were you, I would download the free trial and see whether a newer version is released between now and then. 

Generally, owners of any prior version of Elements have been able to get a discount on the purchase of a new version when it is released.  Then, as we approach the holiday season (Black Friday in the US etc.), other retailers (and even Adobe) give larger discounts than the upgrade pricing.  This is only what has happened in the past.  Who knows whether things will change during the current pandemic.  Your guess is as good as mine.

The Organizer's slideshow can be exported as either 720 or 1080 resolution (and there are no options to increase bitrate or change the codec).  Presumably you have at least an HD resolution TV.  You should see no noticeable loss in quality over simply outputting a photo displayed on your laptop screen to the TV.

There is a 30 day free trial of Elements 2020 which you can find here.  Based on past history, by the time your trial expires, a new version will be out. So, you can determine for yourself whether the program suits your needs. You haven't stated what version of Elements you have now.  If it is more than 3(?) years old, be aware that the newer versions have an entirely new slideshow editor.  Some folk prefer the old one and will keep the older version of the program installed in order to retain the legacy editor.

You can save a slideshow as an mp4 file which you can transfer to your laptop via your home network or a USB thumbdrive.  You can then play the mp4 on your laptop.