How bad are things for Flipkart today (early 2016)? How are they competing with Amazon (on price, service, delivery, etc.)? How has that changed over the last year?

Things are getting worse for Flipkart. Flipkart's honeymoon period is nearing its end. I strongly believe Amazon of India is Amazon, not Flipkart and Uber of India is Uber, not OLA.

Few reasons why coming years will be tough for Flipkart are

1. Money : Money plays a vital role in running any business. Amazon is running profitably across the globe and have enough cash to backup its low performing markets whereas Flipkart is yet to break even and make profit. Till date, Flipkart hasn't make any revenue and losing investors money. The operational charges for Flipkart is getting bigger and bigger on every year. To run a smooth business, Flipkart has to depend on the investor's money where as Amazon depends on it own cash reserves. For example, when Flipkart management has gone through series of meetings to raise the last $1 billion funding round, Amazon very calmly announced its $2 billion investment in India on the very next day. 
2. Users : I have striven from poor to upper middle class. I have good understanding and personal connections with people across all economical backgrounds. Indian market is one of the toughest market to crack for online businesses. Most of the Indians are not brand loyal, they always look for better offers. If someone gives the product in same quality at better price, brand loyalty won't stop us from switching the brand. If someday Flipkart stops offering discounts, many won't bother to switch to Amazon or brick and mortar stores. Till date, mobile phone segments is the top selling products in Flipkart where it provides deep discounts. If they stop giving the discounts, many people won't bother to purchase from the local stores again.  I have seen many experts suggest that the device accessibility, low credit card penetration and internet connectivity are the main issue behind low traction of the e-Commerce stores in India, actually the poor literacy rate and over cautious Indian mindset are the real issues. Apart from this, many Indians are not aware of the e-Commerce app's purchase flow. I have personally know few set of people who have completed their undergrads and seeking for help to place an order in e-Commerce stores though they have android mobile phone, internet connectivity and credit card/debit card/net banking. I really doubt there are millions of people like them are there across India especially in Tier-2 and Tier-3 cities. So, Connectivity and credit card penetration are not the only issue. Flipkart has to go longways to tap this market properly.
3. Technical strength : Flipkart lags behind the Amazon in their technical strength. Amazon has upper edge in technology due to its great experience in the retail sector. Flipkart's inability to scale and serve user surge became evident on its first big billion days. Though Flipkart founders apologised (Flipkart apologizes to customers for poor user experience on its Big Billion day)  for the inconvenience, it is fair enough to show its technical abilities. Where as Amazon is building the great Amazon Web Services for other businesses to scale their business needs without any hassle. Flipkart is fairly lacking behind Amazon in technical strength.
4. Operation expenses : One of the major operational expense is maintaining warehousing and logistics. Amazon is inventing many solutions to solve this issue. Starting from warehouse maintenance robots to drones for delivery, they are deeply investing across many things to reduct the cost. Flipkart couldn't afford all that in its current state. Amazon can fund all these from its excessive cash reserves.
5. Failures : Most of the Flipkart's experiments are failing starting from its tablet, payment solution , Myntra website shutdown and grocery service. These failures are haunting Flipkart management during its fund raising.
6. Traction : After starting its business in India, Amazon manages to become the top e-Commerce website within 2 years (Amazon overtakes Flipkart as most visited Indian e-com site: comScore). Though, Flipkart manages to be the top e-Commerce company among mobile visitors, i highly doubt it is due to its mobile only offers. Once Flipkart ran out of its offers, it will loose that traction too.
7. Unique Selling Point : Flipkart doesn't have any remarkable unique selling point apart from deep discounts on its product listing. Flipkart can't live with that for long time. Unless Flipkart makes a profitable unique selling points for its customers, it won't live long.

If Amazon manages to push hard in the market and make it difficult for Flipkart (especially after 

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Morgan Stanley slashes Flipkart’s valuation by over 25 per cent)  to raise funds in the future, eventually it will get shut down or bought by Amazon in few years.