Very cool design, and definitely invokes a feeling of storm magic.

In terms of abilities, I'm tempted to say that the first +1 is too strong. I know planeswalkers have gained loyalty for dealing damage to creatures, but they've never been able to do so while dealing as much as 3 damage --- the only exception to this is Ugin, who is at 8 mana. So I'm not sure, it's more of a gut feeling for me.

The second +1 also seems to be at a higher power level. Don't underestimate how strong deck and hand manipulation is, though the fact that this does both at once is quite cool.

On the flip side, I think the -X here is rather weak. Even though the flavor is dead on, at 6 mana, requiring 3 colors, and starting at a modest 4 loyalty, I feel like this could easily be changed to "Tap X creatures. If they have flying, Askarn, Planar Tempest deals X damage to each creature tapped this way". At least there's some utility, even if the creatures don't have flying.

The -8 here also seems a little weak to me. While I know "ultimate" abilities aren't what make or break the card, I feel like some templating changes would make this clearer and more appealing. How about making it "Draw a card for each Island you control. Then, put a creature onto the battlefield for each forest you control. Askarn, Planar Tempest deals damage to target creature or player equal to the number of mountains you control".

I don't think that his +1 is too strong. Many other 6 mana planeswalkers have a card advantage as their + ability, but we also have Sorin Markov, who deals 2 damage (also to players) and gives 2 life. And that one wasn't known to be a strong planeswalker. Ugin also deals his damage to players.

I don't even know how strong the second ability is. Considering how late this card comes into play, discarding two cards is not easy, but it is also fair that you dig/draw so many cards. In midrange decks it's neat to get some cards you might need, but you may need land cards you still have in your hand. Control decks are rarely temur coloured, but it would make such a deck actually viable.
Right now I am fine with the power level of that ability.

Btw. I would write "...in any order, then draw two cards." (comma instead of period)

It is probably "allowed" to have a rather weak -X ability, since you have 4 abilities in total and the chance is high that the planeswalker becomes too overpowered then.
I like both your version of the ability and the one GrandTaurus chose (because I like flavor).

His ultimate is underwhelming. What creature card do you still have on your hand that you can't cast anyway? I would think to change it to something like this:

"Search your library for a land card for each forest you control and put them onto the battlefield. Then shuffle your library. Draw a card for each Island you control. Askarn deals damage to target creature or player equal to the number of mountains you control"

Note: Askarn should be the one dealing damage, not the lands in this case I think. But for flavour reasons it doesn't have to be. At least "Each Mountain you control deals 1 damage to target creature or player." is shorter text. They write 1 as a number, when they talk about damage btw.

The ultimate feels more game winning then and you probably draw some creatures cards that you can play with the new lands in play, but I don't know if that even fits in the card box.

btw. the full name of legendary or planeswalker cards is mentioned only once - the first time it appears on the text box. Then it only reads "Askarn".

In the card is well made, very flavorful and fitting the colour wheel.

You make a good point about the first +1, I was thinking about it just in terms of amount of damage to a single creature, not in terms of card advantage.

The second +1 is a little hard to judge, but it is an ability that will always be relevant.