For instance, if BTC starts to fail, would it take LTC with it?
What coins future are tied to BTC?
What other popular Altcoins don't have this issue?
I will focus on those alt-coins that use POW for consensus and whose primary function is their use as money. The only solution I have seen to this issue requires both an adaptive blocksize limit and a tail emission. Bitcoin's problems with blocksize can be traced to the need to create scarcity in the blocksize in order to have fees rise to provide an incentive miners when the emission runs out.
If we go down the Marketcap list with a mineable and no premine filter turned on https://coinmarketcap.com/currencies/views/filter-non-mineable-and-premined/
1. Bitcoin
2. Ethereum N/A
Note: Primary purpose is not use as money. Has a different problem: Currently the price of Ethereum is tied to the price of computing power https://bitcointalksearch.org/topic/m.14924308
3. Litecoin: Same problem as Bitcoin: Fixed blocksize and Fixed maximum number of coins
4. Dash: The POW has the same problem as Bitcoin: Fixed blocksize and Fixed maximum number of coins.
Note: The Masternode network is also dependent on a falling emission curve; however it maybe possible to use the masternode network voting mechanism to set fees.
5. Dogecoin: POW has only one of the BItcoin problems namely a fixed blocksize. Dogecoin has a tail emission so it could be possible to add a Cryptonote style adaptive blocksize limit via a hard fork. (see Monero and Bytecoin below)
6. Steem: N/A
Note: The primary purpose is a social network.
7. Monero: Does not have this issue. This because it uses an adaptive blocksize limit (Cryptonote) and a tail emission. The adaptive blocksize limit uses a penalty function that is based on the emission and fees are set to overcome this penalty in order to scale the blocksize. This works only because there is a minimum tail emission since if there were no emission there would be no penalty and adaptive blocksize limit would fail.
8. FedoraCoin: Could not verify since the links to Fedoraco.in fail. Maybe someone from the Fedoracoin community can provide answers as to blocksize, tail emission etc.
9. Peercoin: Proof of Stake and has the same fixed blocksize issue as Bitcoin. The POS could provide an effective tail emission. (See Dogecoin above)
10. Bytecoin. Has the same type of adaptive blocksize limit as Monero but no tail emission. The problem with this is that as the emission runs out there is no way to secure the coin since the blocksize can increase with no effective penalty.
Note: Bytecoin had a very large premine/ninjamine which will accelerate the time frame when the emission runs out and could provide an early warning of what can go wrong if one uses an adaptive blocksize limit with no tail emission.
Edit: Many of the above coins fail because there were forked from BItcoin or Litecoin and inherited this problem.