If you are going to have it plugged in to AC all the time at home, I would fully charge the battery and take it out and store in a cool, dry place. My original battery stayed in the computer all the time and I used it plugged in most of the time. Didn't take long for the battery to give up the ghost because it was contantly being recharged. A battery is good for only a certain number of charges. So I bought a much larger capacity replacement battery, use it sometimes, and will be using it when we travel. Otherwise, here at home, I take it out of the computer and store it in the box it was shipped in. These instructions were on the website of the vendor where I bought the battery.

Any new laptop you buy will be able to connect to the Internet if you already have an Internet service at your house. If you want to switch from BellSouth to AOL or to any other provider of Internet services, it doesn't matter: the laptop will still be able to connect to the Internet through whatever provider you choose. Most providers do not need to have any special software installed; you just need to have a browser installed, like Internet Explorer, Firefox, Opera, etc., and all of these are free. At least one (probably Internet Explorer) will already be installed on your laptop when you buy it. If your MIL does not have Internet service, you generally can't connect when you are at her house. Think of Internet service like regular, wired phone service. If you are at someone's house and they don't have a phone line, you can't make a phone call. Now I am going to tell yo something a little more complicated. When you have Internet service at your house, the company will typically provide you with a little electronic box called a router. This device hooks up to a cable in the wall that carries the signal the company provides, and then a wire goes from that little box to your computer. You can get access to the Internet from the wire that comes from that router, or you can connect without the wire because the laptop has a card in it that can receive the Internet signal wirelessly (sort of like a cell phone) from the router. That way, you can move around the house easily with the laptop, or maybe even sit outside with it. The disadvantage of a wireless connection is that it has limited range. The computer may not be able to receive a signal in some parts of your house. On the other hand, if the signal was strong, you might even be able to get it wirelessly and connect to the Internet at your MIL's, even though she doesn't have her own service. Given that you know little about computers, probably any laptop will be just fine for you. You should choose one primarily on the basis of the screen size and how much you want to pay. Don't get talked into spending a lot because all the features and speed you need will be present in even the lowest cost computers. If you go shopping and are confused, ask more questions here before buying. Lastly, there are many good books out there that will explain what you need to know in very simple language. Get yourself a book like Computers for Dummies or something similar like the one below. (In trying to answer your questions, I have assumed that whatever Internet service you have that it is not the old-fashioned, painfully slow dial-up that uses a modem. If you do use dial-up to get Internet service, then there is other stuff to tell you. Here is a link that might be useful: Absolute Beginner's Guide

How about trying another site that offers the New York Times Crossword and see if it will print for you. The Seattle Times also offers the New York Times Crossword by Will Shortz, the same fellow who does them on your Arkansas paper. On the Seattle Times page, click on the printer icon. On the next "page" select "puzzle with blank diagram" (it may already be selected). Click "OK" The puzzle should appear. Click on the printer icon. If the puzzle doesn't appear, then you probably don't have a PDF viewer installed on your computer. It's worth a try, however, especially if it works for you. Let us know. Here is a link that might be useful: New York Times Crossword

I have 2 HP Pavilions at my house and do not have the problem you are experiencing. I did have an issue that my battery would drain to very low charge if the computer was stored for a week. However, my issue was more with the "Window10 operating system than the computer. Something was active when the computer seemed to be shut down. A fully charged battery should be able to keep the BIOS settings alive and the clock/calendar running for many days, probably months. I did find one setting that was causing a drain. About slow to not charging: Two items could be at fault. One is the power supply that is in the power cord. One cord plugs into the household electrical outlet and the other end, carrying DC power goes to the computer. The power supply may be faulty. There has been problems with batteries shorting from over charging or trying to charge a failed battery pack. If a portion of the battery pack has shorted, then its output voltage would be diminished and a charger trying to bring that pack up to voltage overcharges the remaining good cells. If it does not fully charge overnight, there is something amiss. The problem could be the battery pack or the power supply. Our HP Pavilion computers come up to near full charge in 2 to 3 hours. By the way, make sure you did not plug it into a switched outlet that goes dead when you turn the lights off.