First off, the champs you have available to you follows a weird system, and there's no breaking it. To start off with, there's a circulation of 10 or so free champs every week, which you don't have to own to play. Once the week ends, that set of 10 stops being free and the next set becomes free. If you want to permanently unlock a champion, you can pay either IP, which you earn for playing tons of games and racking up the wins, although most of the champions are quite cheap so you'll be able to get a pretty decent pool rather quickly. The other option is RP, which is paid currency, but it's definitely not pay-to-win by any means, those are the only thing you can buy with it that have an actual impact on the game. If you choose to take a random champion, it'll select one from your pool of available champs, which are the ones you've permanently unlocked plus whatever the week's freebies are, nothing else.
You won't be able to do that with specific champs (LoL calls em champions no heroes. I don't care about being specific, but some of the dota vs lol fanboys get picky.) for very long. There's the period up until you learn all of the champions where that can be the case, and much later on if you ever get there where you're theory crafting and want to try something new because of a skill change or something.
For example, Taric, who is something like a Paladin, he has a heal and he's supposed to be hard to kill and passively give teammates around him stat bonuses and stuff, recently got a bunch of changes and so a handful of people tried him in other positions in the standard set up than he normally is.
As for constructing tactics every match, yeah, that's a thing. Basically, you have to be really adaptable to figure out what you're going to do in reaction to what goes on each game, both in champ select and 20 minutes in. The times when this gets most crazy is known as ARAM, all random all mid. Basically, it picks a random champion from your pool and you're stuck with it. Then you spend the next half hour duking it out with 9 other people who are just as likely to get someone they play regularly as you are, so half the time you'll get a champ you haven't played in months and be like "Oh wow, I never remembered [Whoever] was so fun to play, this is great!". Other times you'll get your favorite champion and then completely crush the other team. It's finicky like that.