Do I need a tool, or will the chain have the correct length? For example, my biggest cog in the rear is 34T and the biggest chainring at the front is 42T and the chain has 116 links. Is a chain with 116 links enough? What should someone consider about the length when buying a chain?
You need a chain tool. All drivetrains are different. The chainring size (or the big ring size if a 2x or 3x drivetrain), largest cog size, and chainstay length all influence the length of chain needed, and all bicycles differ in these parameters. 116 links is a standard length that suffices for the longest required chains on most bicycles. Hence, that length will provide just enough or too much chain for most bikes.
Naturally, you can outsource that task to your local bike store. Each person's inclination for doing their own work and maintaining the necessary tools is different. Changing your chain is easy enough, and many cyclists should expect to do it often enough, that I think a chain tool is a worthy investment. However, your bike store will be more than happy to check for wear and install a new chain, although their workload may affect how quickly they can do it (NB for people happening on this answer in the future: there is a global pandemic now, and bike stores are working overtime since many people want to get their bikes fixed plus they often have to reduce capacity to let their workers stay at a safe distance, so this is not a trivial issue as I type this. We are, we hope, at the beginning of the end of the pandemic.)
Not necessarily, but having the right tool it is the most secure and comfortable way of shortening the new chain, which is almost always necessary.
Recently I had to change a worn out chain and I wasn't home, I was a someone else's garage and he would lend me his chain tool only to find the tool was damaged and was unable to remove a pin, so no way of using it to cut open the existing chain or shortening the replacement.
Unfortunately, the old chain had no quick link or connector, so it had to be cut with a cable cutter. (That's what I had on hand, A grinder with a cutting wheel would had been handy)
I shortened the new chain pushing the pin out with a hammer and a nail sinker tool. The chain was placed over a bench vise with the jaws slightly open, leaving a small gap for the pin to fall into. (In the past I had performed this task with a blunt nail and placing the chain over anything with a small hole or gap for the pin to fall in).
Once the chain was shortened to the right amount of links, it could be installed with a quick connector. (Included in the package of the replacement chain)
If I hadn't had a connector, the task of re-joining the new chain would have been much, much harder, but doable. (I had done it in the past with large pliers)
So, the task can be accomplished but it is much more difficult and you must be very careful not to damage the chain (at least the part your'e going to install).
(In my case, I learned my lesson: If I offer myself to perform maintenance while on someone else's home, I'll carry my own tools)