I am composing a piece for full orchestra and want to make sure that my parts are conventional. I know that there are typically four horn parts, but are 1 and 2 high parts, 1 and 3, or something else?
Conventionally, 1 and 3 take higher parts and 2 and 4 take lower parts. In many scores, horns 1&3 are given one staff and then horns 2&4 are given the next staff. I prefer that method, but you will see plenty of scores that group 1&2 and 3&4—however it will still generally be the case that the odd numbers have higher parts and the evens have lower. Of course, when you make parts, all of this should be separated out, and there should be individual parts for each horn.
The reason is historical: horns 1&2 used to be in a different transposition that 3&4, and composers tended to have a high and a low horn in each transposition. For example, horns 1 and 2 might be in F, and 1 would take higher notes than 2, while horns 3 and 4 might be in G, with 3 taking the higher notes. Modern horns are always in F, but 1&3 taking higher parts than 2&4 has stuck around.
1 high, 2 low - and you can add 3/4 and 5/6, even 7/8 (e.g. Mahler), but always high/low, always in pairs. Solos often go to 1, but can also be covered by 3/5/7 etc.
It might help to think of each pair of horns as a separate group, like pair of woodwind instruments. You can just add more pairs and occasionally have them all play together. Often the 3/4 horns are doing something completely different in rhythm or key. It's also perfectly OK to use only 2 horns.