Is Monero intended to be a competitor to Bitcoin, or is the intent to develop technology that will improve Bitcoin? Is there any reason why Monero's features wouldn't just be brought to Bitcoin through sidechains? What about Monero's features being implemented in Ethereum?
A lot of the other answers address some but not all of the questions.
Is Monero intended to be a competitor to Bitcoin, or is the intent to develop technology that will improve Bitcoin?
It is unknown whether Bitcoin will ever implement any of Monero's technologies, but from the outset, the cryptonote protocol (from which Monero has evolved) was made to address some of the shortcomings of the bitcoin protocol. From the cryptonote whitepaper, each of these drawbacks is listed - 2.1 Traceability of Transactions, 2.2 Proof of Work Function, 2.3 Irregular Emission, 2.4 Hardcoded Constants, and 2.5 Bulky Scripts. Additionally, from the whitepaper, we see this passage -
For example, the system’s distributed nature is inflexible, preventing the implementation of new features until almost all of the network users update their clients. (...) In such inflexible models, it is more efficient to roll-out a new project rather than perpetually fix the original project. In this paper, we study and propose solutions to the main deficiencies of Bitcoin. We believe that a system taking into account the solutions we propose will lead to a healthy competition among different electronic cash systems.
Therefore, I can only conclude that the original Cryptonote protocol was made to be in direct competition to Bitcoin (to stand alongside) and was not intended to ever be merged into the bitcoin protocol, but only time will tell.
Is there any reason why Monero's features wouldn't just be brought to Bitcoin through sidechains?
There's plenty of reasons for bitcoin to bring Monero's features into Bitcoin through a sidechain, but there's no reason for Monero to become a Bitcoin sidechain. If we take all of the features of Monero and bring them as a sidechain, this sidechain will have dynamic block sizes, an ASIC resistant proof of work, and last but not least, untraceable transactions. Albeit these features, this theoretical Monero sidechain would now be coupled to Bitcoin's security model and other parameters. In addition, as others have mentioned, one could trace transactions being moved to the Monero sidechain and this would be the equivalent of getting $10,000 in cash. You would be suspected of doing something.
What about Monero's features being implemented in Ethereum?
This is similar to the Bitcoin implementation.
The bottom line is that if true fungibility isn't in the base protocol, then the base protocol isn't fungible no matter how many layers you add.
I think the answer is both
Aside from the privacy advantage of not having worry about being identified as a ring signatures side chain user, Monero offers other advantages such as a hedge of having an alternative hashing algorithm to secure the network. In addition, next level advancements such as RingCT and Kovri will likely come much later in a ring signature sidechain (if they come at all).
Bitcoin holders see the value in privacy and do have the strongest network hashrate (by far). Even with the concessions above, I think it is very likely that a ring signature side chain will come at some point.
The impact of #2 is hard to evaluate. On one hand some BTC users will stick with BTC instead of Monero is sidechain privacy is "good enough". On the other hand since the Bitcoin community is many times larger than the Monero community (today) the embracing of ring signatures by a Bitcoin sidechain very likely would increase interest in Monero itself and lead to far greater Monero adoption despite the existence of a BTC sidechain.
From my understanding, the problem with implementing ring signatures as a Bitcoin sidechain is that you would stand out like a sore thumb if you transfer normal Bitcoin into the ring-sig sidechain. The actual transactions on the sidechain might not be traceable but the transactions to/from said sidechain may well come under additional scrutiny (e.g. by Coinbase who currently flag and disable accounts that have transactions to known gambling sites).
With Monero, ring-signatures are baked into the protocol so everyone has to use them. So that helps to protect the privacy of everyone who uses Monero, rather than just a (relatively) small, potentially visible subset of people who use a Monero-esque sidechain on Bitcoin.
As for Ethereum, I don't know enough about it, other than that it's a "Turing-complete" platform, to be able to say whether ring-sigs could be properly implemented. Regardless, it would be a small subset of Ethereum users that would likely take advantage of any ring-sig capability and thus transactions to/from ring-signature Ethereum implementations would likely be just as visible as if it were a Bitcoin sidechain.
Including it as a sidechain would not make Bitcoin transactions very private as you'd still know which coins were used in the sidechain. Ring signatures would need to be baked into the Bitcoin protocol, and become mandatory for Bitcoin to enjoy the same level of privacy and the chances of this happening are probably extremely slim. This would be the same with Ethereum, and Vitalik himself has said that protocol level changes of this nature simply aren't going to happen. This makes Monero a standout among other cryptocurrencies as it is prety much the only major blockchain in existence today that strives to make a fully private cryptocurrency. Zerocoin MAY be an alternative, but it is still a long way off and is based on very experimental crypto whereas Monero's built on extremely solid cryptographic principles.
I find Monero is not necessarily a competitor, but more a different tool for different purposes. They overlap, but are also distinct enough that they could exist side by side.