Connecting an HDMI monitor to a VGA laptop without losing resolution

It is possible, but not trivial. You will need an actual active converter rather than a cheap cable. The reason for this is that HDMI is digital only, and the old VGA signal is analog.

As to the quality, that depends on the output on your VGA/DB15 plug and on the quality of the convertor. I looked before and I could not find any limitation in the VGA standard. However most VGA cards come woth a 400MHz DAC which would commonly limit it to any configuration using a lower than 400MHz bandwidth. (E.g. 2048x1536@85Hz which needs 388 MHz bandwidth). If your DAC is less capable or there are other limits then the maximum resolution will be lower.

if I use a HDMI converter on the VGA cable and connect to my TV, will I get widescreen resolution?

No. Not automatically, and/or if the problem is the fact that you are using VGA in the first place.

This is simply because a converter cannot introduce more data than is already there; it can only convert the data that is already there to some other form. For example, a VGA input to a HDMI output. The adapter would have no way of knowing what to display on the additional 281,088 pixels.

However, VGA is perfectly capable of supporting the resolution you want at reasonable refresh rates. Something else is wrong with your setup.

I would recommend starting out with installing the latest drivers for your graphics card from the manufacturer.