Tell me please, do you need to have coins on your local wallet to receive Free Weekly Airdrop, or can they be held on the exchange?

You go to https://www.bitcore.cc/ and follow the steps.
You'll have to sign a message from each of those wallets. The message is the same, iirc it's BTX. You'll need a BTX address too, you can use cryptopia.

Is there a time limit until I can make that claim? Or is the overall amount that can be claimed limited?

My mistake huh? Obviously this board is full of idiots. No mistake on my part I assure you. Ive made over 300 withdrawals over 8 different accounts. And this is the first time I got Fucked. So where are my coins? They went to somebody obviously. I thought everything could be traced on the blockchain??  But yet nobody can help me??!!  Lmao.  Geniuses here!!  And check back about 20 hours I posted about a new coin on C-Cex.  BITCF.  I said buy.   Well its up 200% now on Coinmarketcap and C-cex.   I made a quick triple.  So keep holding your dumb BTX.  Im glad now that I got screwed. I already sent what I had left back to Cryptopia and sold it.,     DEAD MONEY>>

But if any of you people want to make some real money, PM me.  I started 5 months ago in Altcoins, and now im up 5000%.  And I knew nothing when I started.  Ill even give you a couple free picks.  Or just keep holding your BitBore....ing

Oh my bad. BITCF is only up 147% in last 24 hours now. darn.  It launched last night at a starting price of .0136$  it got as high as .0604.  now its at .0566.   i have a 2 week target price of .180    So you can still triple your money in 2 weeks.  Or not. I could care less.

And to be perfectly honest. I dont think BTX is a bad coin. Obviously I wouldnt have bought it if I did. But they definitely need some help in the customer service area.  You dont get people interested in your coin by ignoring them and screwing them over. I bet if i woulda been a whale and was looking to buy 50,000 BTX at $2.75 they would find my frikin deposit for me. lol

Nothing annoying about someone making an alt to quote themselves about a price and ignoring the factual statement provided against their information/short-comings when using a wallet.