Tripod (or even monopod) would have absolutely helped. If your lens /camera doesn't have any type of compensation for shakiness, you're bound for trouble unless you're one of the very few folks who happen to have super steady hands. Some can train for that with enough practice, others can't. The more you zoom in somewhere, the worse things get. If you zoomed in with that zoom lens of yours, you're almost guaranteed camera shake, at any setting. So yes, for shots like this you very likely need some kind of physical stabilization - tripod, monopod, or even a chair you can rest your lens on.

You can install Lightroom on any computer that runs either Windows, or Apple OS (Mac). Laptop or Desktop doesn't matter, those are just form factors. Lightroom does require fairly decent resources. A lowest-end type of computer may not provide you with a smooth experience. But then, no photo processing software will.

In your particular situation I actually recommend Lightroom as opposed to Photoshop. LR is more geared toward the photographer. It's easier to make batch adjustments, create profiles, do before-and-after comparisons, and is just generally more geared toward adjusting photos. Photoshop is a photo MANIPULATION software that is way overkill for most people. It's what you use to make noses smaller, insert your dog into a photo, remove the background, and so forth. If you don't plan on doing significant manipulations like that, LR will be your better bet. LR is also made by Adobe (like Photoshop) and because it's become quite the go-to application for so many photogs in a fairly short time, I'm willing to bet their support and upgrades are going to continue for a very long time.

Been using third party batteries for my personal as well as my business' and my clients' equipment for years, generally without issue. We use them on our old 7D but also G30s, G40s, as well as for non-Canon brands (an arsenal of Sony gear and Sony-compatible battery devices such as LED lights). There is indeed a difference between the manufacturers, and as pointed out here before, Wasabi tends to have a pretty decent reputation. I also use POWWER LP-E6, which actually has a higher 2600 mAh rating compared to Canon's own 1800 (have not noticed a real world difference however, not even in burst).
I also have good experiences with Digital NC and Kastar.

It's true that some third party makers don't last quite as long as the OEM Canons, but once again, your mileage may vary. In our day to day operations, the difference is very minimal - or even not noticeable. For the significantly lower cost, you can buy a bunch and keep them around in case you do run out of juice a bit early. At no point did we ever experience a significant issue such as only getting half the photos or video compared to OEM Canon. I would estimate that at MOST, the difference was about 20% less performance, and those were the cheapest ones (don't even recall the manufacturers name).

I highly recommend you check out the reviews on Amazon for each one you are considering. If it says "Verified Purchase", you can be pretty sure it's a legit review.