You are on the right track as far as trying to match up the coordinate systems of all of your datasets.  The problem you are encountering is not so much a problem of the software projecting the data incorrectly, but of using the wrong operation to define the projection of your data.

This answer directly discusses the difference between the Define and Project commands in ArcGIS.  It also has links to the relevant ESRI help files discussing the different commands.

Let's start with what factors you are able to control.


You have a shapefile that defines your roads around Lake Tahoe, in UTM Zone 10 NAD 27, from the USGS.  Generally, spatial data from there is going to have the projection already defined, meaning that when you look at the properties in ArcMap, it will list the Coordinate System.  You can use ArcGIS to Reproject this layer as necessary, in this case, to the UTM Zone 10 NAD 83 Meters, projection.  
You have a .gpx file of GPS coordinates that defines a bike trip.  You have the ability to convert this to a file of coordinates in either Lat/Lon or UTM.  This file could then be brought into ArcGIS and reprojected as necessary.  
You know the projection of the Lidar data from opentopography.org.  It is UTM Zone 10N NAD 83 Meters.


The one thing you are unable to change, because of the format, is the Lidar data, which you are going to read using the LidarViewer software.  You thus, must ensure that the coordinate systems of all the other data layers, matche the Lidar, since you will be importing the other data into the Lidarviewer as well, to overlay.

Here is the process.


Look at the Roads shapefile.  In ArcMap, if you look at the properties, and shows that it has a coordinate system defined, you will need to use the Project command to Reproject the data from UTM 10 NAD 27, to UTM 10 NAD 83.  
a.  Convert the .gpx data using the website, and save as Lat/Lon.  This should give you a text file of coordinates in degrees.
b.  Add this using the Add X/Y Data command in ArcGIS, specifying the coordinate system as:  Geographic:  NAD 83.
c.  Use the Project tool on this new layer, and choose the UTM Zone 10, NAD 83 Meters coordinate system as your output.


What you should end up with is all 3 input layers being in the same coordinate system.  You can now import them into the LidarViewer and they should overlay properly.

A note about the LidarViewer software:  Even though it does not support coordinate systems directly, it will still overlay layers properly if they are all in the same coordinate system to begin with.  This simply means that all of the projection has to be performed prior to the data import.

Aligning coordinate systems: gpx, LiDAR, shapefile
Polyploidy is the multiplication of number of chromosomal sets from 2n to 3n (triploidy), 4n (tetraploidy) and so on. It is quite common in plants, for example many crops like wheat or Brassica forms. It seems to be rarer in animals but still it is present among some amphibian species like Xenopus.

As I know in mammals polyploidy is lethal (I don't mean tissue - limited polyploidy). I understand that triploidy is harmful due to stronger influence of maternal or paternal epigenetic traits that cause abnormal development of placenta, but why there is no tetraploid mammals?
