Are tubes really “warm”?

Usually yes but the degree to which varies widely. 

It depends on the design and the company ARC not to warm CJ warmer etc.

What I really found out in tube amps is you can never get great sound of Strat with solid state amps. The sound won't be warm at all. 

It depends on the amp and the speakers, and the era.

The way to design speakers in the top octaves has varied a lot over a few decades, from a rolloff starting around 7kHz, to flat, to ragged and exaggerated.

There were a number of amps which in the past would sound dead cold to me, old Audio Research, but which now sound more neutral and pleasant.

Not necessarily. I’ve heard certain tube circuits and/or certain tubes that could peel paint. On the flip side, some badly designed/implemented tube amps can have massive roll-off and gobs of THD, and that can be used to perpetuate the stereotype of "warm, rich tube sound", but it doesn’t have to be that way. Certain tube headphone amps and cheap Chinese tube amps can be particularly guilty of either kind of bad tube sound - but is that stuff really a proper "tube amp" or should it be thought of more as a novelty? Really good tube gear should be reasonably neutral with THD at least somewhat under control, and featuring good extension on both ends.