H.264 and MPEG-1 codecs are both built into Vegas. You don't need to download anything for those. In general you don't need Quicktime to play them. You usually don't even need Quicktime to decode H.264 in a MOV wrapper. It will be decoded by Vegas' own codec. Note that H.264 is the same as AVC. Why do you want an MPEG-1 codec? It's a very very old format.

H.264 and MPEG-1 codecs are both built into Vegas. You don't need to download anything for those. In general you don't need Quicktime to play them. You usually don't even need Quicktime to decode H.264 in a MOV wrapper. It will be decoded by Vegas' own codec. Note that H.264 is the same as AVC. Why do you want an MPEG-1 codec? It's a very very old format.

My vegas 10 doesn't have H.264 even though I'm able to choose H.264 in Bandicut

You'll have to forgive me, but I'm a newbie when it comes to codecs.

What codecs is everyone using today?

If MPEG-1 is an old format, then what is the latest MPEG codec?

Is there a drastic difference in quality and file size between the old and new MPEG codec?

The H.264 codec is the most common codec in consumer video! As long as you've installed Quicktime 7, per the program's requirements, you should have no problems editing it as well as a number of other major formats. (You can remove or disable the Quicktime Player, for security reasons, but you will need the Quicktime components.)

My video converter lets me select H.264 as codec but my Vegas 10 is unable to read/recognize it? Do you know what the issue is?

Perhaps if you told us what video converter you're using, someone might be able to help.

Please use the program MediaInfo (a free download) to display the particulars of the H.264 file, and post the output here.

The video converter is Bandicut

From MediaInfo

Format                                   : AVI
Format/Info                              : Audio Video Interleave
File size                                : 9.42 MiB
Duration                                 : 31 s 300 ms
Overall bit rate                         : 2 526 kb/s
Writing application                      : Bandicut 2.6 (Lavf57.41.100)

Video
ID                                       : 0
Format                                   : AVC
Format/Info                              : Advanced Video Codec
Format profile                           : Main@L3.1
Format settings, CABAC                   : Yes
Format settings, ReFrames                : 1 frame
Format settings, GOP                     : M=1, N=12
Codec ID                                 : H264
Duration                                 : 31 s 300 ms
Bit rate                                 : 962 kb/s
Width                                    : 1 024 pixels
Height                                   : 768 pixels
Display aspect ratio                     : 4:3
Frame rate                               : 30.000 FPS
Color space                              : YUV
Chroma subsampling                       : 4:2:0
Bit depth                                : 8 bits
Scan type                                : Progressive
Bits/(Pixel*Frame)           

Bandicam footage generally causes problems in Vegas, so presumably Bandicut might too. AVC video in an AVI file (which your MediaInfo report shows) is unconventional. AVC video in an MP4 file would more likely be supported natively by Vegas if Bandicut can create that instead. In general you don't need to download codecs to use Vegas but in this case the x264vfw codec might help you play those files. I would download that from here: http://www.videohelp.com/software/x264-VFW If it doesn't help then uninstall it again.

As background, AVC is H.264. Same thing. x264 is an open source codec that can encode H.264 video. vfw is Video For Windows, a framework that uses AVI as the file extension.

For screen recording, OBS Studio generally records files that Vegas likes.