RP skins reward Your best bet would be to input the other player's username in the Gamekit ordering screen instead of your own League of Legends username. It obviously wouldn't be a surprise to them after that, but there is no way to "gift" a received skin without being the person purchasing the skin initially.
Skin Ordering Process (It's legit) @aresbackup423: Heartseeker Lucian isn't available to buy yet. Typically the Heartseeker skins are available for a limited time around Valentine's Day in the US (February 14). If you check the shop, there are only 2 Lucian skins available for purchase at the moment: PROJECT, and Hired Gun.
Cancelling LoL skin order Well if you've accepted it, then that means they can send you the skin at any time and if they did, they would be unable to cancel/refund your order since, as far as I'm aware, there is no way to return a gifted skin to the original sender. So if you've accepted, then it's kind of a race against time between how soon your ticket gets answered and your order canceled by a mod, and them gifting you the skin.
Cancelling LoL skin order You might be able to cancel if you go into your order history. If you can't, then send in a ticket asking to refund the order and get your points back. If they send you a friend request, it may help to not accept it so they can't just go ahead and send you the skin you ordered.
Screenshot I just used Windows Snipping Tool, but here you go, from their Wiki. "To take a screen shot with the User Interface, use the command /screenshot_ui (for TGA) or /screenshot_ui_jpg (for JPG) in the chat box."
So basically I've entered some giveaways and realized no one else cares about the reward. Leaving me in a bunch of giveaways that remain open I've never joined a giveaway (never thought they were worth it tbh) but my guess is you just sit tight until it fills up. Next time, I'd recommend keeping an eye on the giveaway when it rolls over to a new one and noting how fast or slow it fills up. Avoid the slow ones and join the fast ones. Or just get the reward directly from the rewards shop if you can.
Super late orders and barely any mod responses Sometimes it takes a while. I've seen people waiting as long as a month. It may be the holidays left people with more free time to earn points and now everybody is cashing those in at the same time and bogging down the system. Give it time, and be sure to log in to League to accept their friend request when you do finally get it. For clarity, they can't gift you anything without first being your friend for 24 hours. Be sure you didn't accidentally decline the friend request.
Not receiving my points? They don't expect you to hit level 5 with exactly 0 experience on the bar. As long as you're level 5, they don't care how far along you are toward the next level. Your experience bar can be halfway to the next level or 10 exp away. Doesn't matter. Just as long as your profile in game shows level 5.
Level Requirement When your screenshot is declined, it typically tells you what the reason was. If you sent in a screenshot when you hit level 5, and it was a full screen, unmodified screenshot including your character's name (the one you entered into Gamekit upon beginning the quests) visible on screen then there shouldn't be any reason you were declined. If that was the case, resubmit the screenshot. Otherwise, you do need to be the exact level specified for that particular quest. If it's 5, you need to be level 5 in the screen shot in order for it to be accepted. I usually won't level past the level required for the quest on Gamekit until I've been approved. That way I know I won't ever be too high a level if ever the case comes where I need to retake a screenshot.
Gamekit is absolute BS Gamekit is legit. Like others have mentioned, you probably just fell through the cracks. Send a ticket and keep up with it. Let them know what happened. It'll get resolved and you'll get your skins. I've never been shafted a skin or RP from Gamekit before and I've ordered around 10.
Skins less than 750 rp? I don't think they use bots to purchase and gift skins. Before they overhauled the ordering process/system, I would receive emails saying that the friend request has been sent. Those emails were separate from the automated Gamekit ones and were clearly written by actual humans (they didn't capitalize, the spelling was off, or I received an email that simply said "added"). It could be different now. However, if they did fulfill the 520 skin order in place of your purchased 750 RP Skin, then you would definitely lose those points. I know some people have asked if they'll refund you the difference or rollover the balance or whatever - they don't. But if you're willing to take the risk and don't care about the extra loss in points, then go for it. At the very least, you'll end up having your order declined and have to send a ticket in to either cancel/refund the points or just change out the skin for a 750 RP skin you might want instead.
Buying skins after first one for league They definitely don't delete you, I still have all the accounts from previous skins added. I wouldn't delete them either, because even though Joel is right that it's super unlikely you'll get the same account the next time, I did. It was pretty convenient not having to wait the 24 hours and just logging in to see I was gifted a skin without having to accept a friend request because I already had some of the Gamekit accounts added.