Hi there, welcome to the site.
Let me break down some of these topics for you...
Support via Chat:
It's primarily budget-end customers that like to use LiveChat, but not because it's actually a useful support model. These users tend to be impatient, self-important, and generally more aggressive in their interactions with the host. These low-paying customers want instant gratification, as if hosting was cooking a dinner in the microwave. When it comes to highly technical services, which hosting is, issues often take time to research and resolve.
Hosts also tend to have two or three levels of support (L1, L2, L3), with front-line techs handling new tickets, in order to resolve easy issues. L2-L3 techs are generally well-trained and can command higher salaries, so it's unwise to waste their time. L1 tech spend most of their time helping clients who fail to understand how the hosting panels work, etc -- and most of that could have been avoided if the client were to read the host's knowledgebase (help documents).
LiveChat techs are often staffed by L1 techs -- or even what could be called L0 support, or people with even less server permissions than L1 techs. The L0 support is primarily there to assist in pre-sales questions or in handling low-level questions. Most technical questions are them simply reading you the answer from their own help docs that are found on their website. But anything outside the scope of basic knowledge has to be "escalated", meaning either your or they will submit a support ticket, and the LiveChat session is over.
While-you-wait support is simply not plausible in the hosting industry. All it tends to do is frustrate customers because of its ineffectiveness, and then it costs the host more to staff it. Those costs are passed on to you, either as outright higher hosting prices, or by means of cutting corners elsewhere in the operation -- the last of which creates a vicious circle, because problems are more likely, thus causing more support chats/tickets!
If you really need tech support -- use the host's ticket system!
Commerce Sites vs Shared Hosting:
I would not run any serious e-commerce endeavor on a shared hosting account, unless it was one of the premium type shared accounts. Those are generally referred to as "semi-dedicated", "business", or "enterprise" hosting. You'll get higher bandwidth and higher disk space. But more importantly, you'll get higher allow concurrent MySQL connections, RAM and CPU time. Those last three are far more important than disk/bandwidth.
-- Stablehost has enterprise hosting, and I highly suggest them.
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MDDHosting has semi-dedicated hosting, and I suggest them as well.
In terms of your connectivity to Singapore or Asia, Stablehost is on the west coast of North America, in Phoenix, Arizona, USA. (Yeah, it's technically not "the coast", but it's close enough to be considered a west coast host. The only better location would be Los Angeles or Seattle, but small network speed increases can be easily undone at a lesser host. Plus latency/ping speeds are not the same as site loading speeds.)
Sales ticket not replied to quickly ?
Some companies give preferential treatment to new customers. New customers get the best deals, quick answers to pre-sales questions, etc. Tread carefully. Sometimes that means they're more interested in new customers, as opposed to caring for their existing ones. Cable companies, satellite TV providers, and mobile/cell phone providers are near-universally despised for this behavior.
Hawkhost is an excellent host, known for their customer service. So if they've not yet answered a sales ticket, it means they were spending time helping existing customers. As a customer, I very much prefer that!
Downtownhost (or subsidiary Ninjalion) good?
We used to recommend DTH and NL, having had good experiences ourselves. Ninjalion always had slow support, which was fine considering the $2/monthly cost and the fact that uptime was near 100%. But hosts are tested when there are issues, and while DTH/NL had done well in past years, their recent non-communication with customers is inexcusable. I can understand downtime -- even days of downtime -- but not being ignored. Some of their servers were down for up to an entire week, with no answers to support tickets.
At this time, I'd say to avoid them.
They've dug themselves a huge hole, and it will probably take a couple of years to recover their reputation, if at all. Even daily mass emails to customers would have sufficed.
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Did I miss anything?