You probably missed talking to Raven after the scene with her and Dolce. You should get a surprisingly long conversation with her about the nature of family. (I didn't immediately realize that the conversation was part of the event, until I started wondering why Raven was so uncharacteristically talkative.) If it's not Raven then it'll be someone else you need to talk to, so talk to everyone.

All you have to do, like in most games, is choose the old save file when you're saving a new one. It's the same for all RF games.

What you can do is busy yourself with preparing for Arcs 2 and 3. Level your crafting skills and work on unlocking all the crops. A little investment in these fields can go a long way in Arcs 2 and 3. Don't forget to explore, there are a few secret areas that you may or may not have found yet. One of these secret areas is the Demon's Den. It is only accessible in the first week of each season. Beating the boss here will drop a Dungeon Seed, which allows you to grow your own dungeons. Delving into these Plant Dungeons may earn you a Sword or Shield Seed, which you can use to grow Plant Swords and Plant Shields respectively. The stats on these items are completely randomized so you may end up with a Plant Sword with post-post-game stats far earlier than is otherwise accessible.

You can buy one from blossom when you get the feature, it will cost you some gold and a few materials. If you are just begining the game I think is best to restart and learn of your mistakes

You have to get their friendship level to 4. The simple (albeit tedious) way to do this is brush them and talk to them every day. That ensures they're in no danger. That will take some time to get them to the required level of friendship. You can also take them adventuring with you. They will get little bits of friendship for helping you, and that has no friendship requirement, so you can get them, and take them adventuring with you right away. After you get them to the correct point, you can talk to them, choose "Assign Field Chores", and then choose Left, Middle, or Right. By doing so, they'll water their assigned section, leaving the harvesting to you.

For regular monsters, the most important thing is persistence. While they tend to like specific items, usually you can use whatever you want, and they'll eventually decide youre alright. To increase your chances, you can brush them beforehand, and then start chucking items at them. Using an item they like will also help your chances, but requires more work to set up. I generally just start giving them spare food when they're close to dead. You can also sometimes just luck out and they want to join you after you kill them. There's a pretty good list right here of what each monster likes. Pay special attention to what boss monsters want; you can tame them, but they generally only have an item or two you can use for taming. They're also riskier to attempt to tame, since they are, after all, bosses. Note that you can't tame a boss the first time you run into them; you have to go back after beating them the first time.

If the character has a relationship level of at least 7 and is a candidate for dating, you will begin dating them. You can see which characters are suitable for dating by looking at the relationship panel. Those suitable for dating will have 1-10 hearts denoting your relationship level, everyone else will have stars. If you say "I love you" to someone who is not a valid candidate (too young, old, wrong gender, etc), not around level 7 relationship, or just a bad day (possibly some unknown reason) they will just take it as a joke. There is no option for a homosexual relationship.

The simplest way is just to talk with them everyday. Its not worth much, but it's reliable. You can also give them home cooked foods. You'll get a pretty decent chunk of friendship from that, and I have yet to meet anyone not like any random food I made.

In your journal, you can see what requests you currently have. When looking at the details of the request, you should have an option to delete/abandon the request.