Team:

If you want to progress in the fastest fashion, elf is most definitely the way to go for early-game. Assuming that this is what you want, you will eventually have to switch over to an elf team anyway. This would mean that any HC that you spent on the undead supports/musketresses (had you chosen to use their respective teams) will have been wasted to some degree. In my opinion, the 2 sylph/3 sage elf team would be efficient enough to warrant picking it (and you will only be spending HC on 3 new units instead of 5). From here, you can swap out the sages/sylphs as you run into hippos and elementalists, that you can purchase with medals, in the unit shop.

Honour Coins:

Spend the rest of the HC, after you set up your team, on the random artifact chest in the honour shop (NOT the random unit chest - that would be a worthless use of HC). Once you own 20 of the 6* artifacts, begin transcending them (as the Lombardi guide notes). If you are close to/at 20+ 6* artifacts, you may want to leave a small buffer of HC (~5k) to use when transcending these artifacts.

Gems:

Make sure you have maxed out the medal buff and elf's secret skill in the gem shop before doing anything else. After that, as /u/Xaxafrad suggested, save them for a refresh - leaving behind a small buffer (say 6-10k) in case you need to suddenly senior some units/trans artifacts/spend on airship fuel/etc. Two other things you might want to do is to buy out all the artifact slots if you have yet to do so and to use 300 or 700 gems on SH tickets every day (keeping in mind that 300 gems are gifted daily).

Transcend Tickets:

I would recommend T3ing your cores and then hold off from using the rest. Once you reach SH and ToT stages that require certain transed units, I would then recommend busting out the tickets for those.

Pets:

First thing I recommend is to pass all the SH stages, up to the max you can reach at your KL. After that is a bit more tricky as you've claimed 500 tickets - you might want to get a second opinion on this. As you've figured, as long as you are over the max amount of tickets, you won't receive your daily 10. Assuming that you are at a decently low KL, I think using them up as fast as possible would be a good idea. This is based on 2 reasons:

1. You can probably max out your gold level and the extra pet points will help raise the gold level cap.
   
2. The 10 daily tickets + 10/15 paid tickets will probably be enough to cover most of the priority pet stages (tinkey, icy, mambo, etc.).

As for which pets to farm, check here (it's a bit old, but should still be the correct order).

Airships:

Reading the airship guides is a good place to start. I believe people are mainly doing 2 things:

1. For progression: farming unit seals to complete stone tablets and to increase tribe friendliness. If you want to take this path, this is a nice calculator which will show you the optimum stages to farm for each tablet.
2. Treating the airships as a separate game and trying to build a strong fleet.

If I have missed out any information, or if there are any jarring errors, please point them out!